Geometric Realization of Triality via Octonionic Vector Fields

Abstract

In this paper, we investigate the geometric realization of $\mathrm{Spin}\left(8\right)$ triality through vector fields on the octonionic algebra $\mathbb{O}$. The triality automorphism group of $\mathrm{Spin}\left(8\right)$, isomorphic to ${\mathfrak{S}}_3$, cyclically permutes the three inequivalent 8-dimensional representations: the vector representation V and the spinor representations $S_{+}$ and $S_{-}$. While triality automorphisms are well known through representation theory, their concrete geometric realization as flows on octonionic space has remained unexplored. We construct explicit smooth vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ on $\mathbb{O}\cong {\mathbb{R}}^8$ whose flows generate infinitesimal triality transformations. These vector fields have a linear structure arising from skew-symmetric matrices that implement simultaneous rotations in three orthogonal coordinate planes, providing the first elementary geometric description of triality symmetry. The main results establish that these vector fields preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$, demonstrating fundamental compatibility between geometric flows and octonionic algebra. We prove that the standard Euclidean metric on $\mathbb{O}$ is triality-invariant and classify all triality-invariant Riemannian metrics as conformal to the Euclidean metric with a conformal factor depending only on the isotonic norm. This classification employs Schur’s lemma applied to the irreducible $\mathrm{Spin}\left(8\right)$ action, revealing the rigidity imposed by triality symmetry. We provide a complete classification of triality-symmetric minimal surfaces, showing they are locally isometric to totally geodesic planes, surfaces of revolution about triality-fixed axes, or surfaces generated by triality orbits of geodesic curves. This trichotomy reflects the threefold triality symmetry and establishes correspondence between representation-theoretic decomposition and geometric surface types. For complete surfaces with finite total curvature, we establish global classification and develop explicit Weierstrass-type representations adapted to triality symmetry.

1. Introduction

The exceptional Lie group $\mathrm{Spin}\left(8\right)$ occupies a remarkable, unique position among classical Lie groups due to its triality symmetry, first discovered by Cartan in his classification of simple Lie algebras [1]. Unlike other spin groups, $\mathrm{Spin}\left(8\right)$ admits an outer automorphism group isomorphic to the symmetric group ${\mathfrak{S}}_3$, which acts by cyclically permuting the three inequivalent 8-dimensional irreducible representations: the vector representation and the two chiral spinor representations. This phenomenon, known as triality, is connected to multiple areas of mathematics and theoretical physics, including the theories of Jordan algebras and exceptional groups, as well as string theory and supersymmetric field theories.

The algebraic framework of triality lies in the connection between $\mathrm{Spin}\left(8\right)$ and the division algebra of octonions $\mathbb{O}$, the unique 8-dimensional normed division algebra over the real numbers. This relationship was extensively developed by Adams [2], who showed how the automorphism group $G_2$ of the octonions embeds naturally into $\mathrm{Spin}\left(8\right)$, and by Baez [3], who connected octonions to the understanding of exceptional structures. Indeed, triality emerges from the non-associativity of octonionic multiplication, which breaks the symmetry between the three 8-dimensional representations of $\mathrm{Spin}\left(8\right)$.

From a geometric perspective, triality has been extensively studied in various contexts, including calibrated geometries, special holonomy manifolds, and moduli spaces of bundles. Thus, Harvey and Lawson’s work on calibrated submanifolds [4] established the role of $\mathrm{Spin}\left(7\right)$ and $G_2$ holonomy in eight and seven dimensions, respectively, while Joyce’s construction of compact manifolds with these exceptional holonomies [5,6] discussed the geometric implications of triality. More recently, the interplay between triality and associative and coassociative calibrations in $G_2$ manifolds, and its connections to the geometry of exceptional holonomy, have been studied by Karigiannis and Lotay [7]. In addition, the triality automorphism acts on the moduli space of principal $\mathrm{Spin}\left(8\right)$-bundles over a curve, leading to an order-3 symmetry that fixes $G_2$-bundles [8] (here, $G_2$ is viewed as the subgroup of $\mathrm{Spin}\left(8\right)$ of fixed points of an automorphism of $\mathrm{Spin}\left(8\right)$ representing triality). The analysis of the triality symmetry in this context leads to a better understanding of the geometry of the moduli space [9], and represents a unique phenomenon, exclusive of the group $\mathrm{Spin}\left(8\right)$, which yields higher-order symmetries, within the line of study of the moduli space involutions that arise in symplectic bundles [10] and in other principal G-bundles in which the structure group admits an outer involution [11,12].

These symmetries provide powerful constraints that simplify the minimal surface PDEs. Indeed, in a triality-symmetric setting, the invariance under the action of ${\mathfrak{S}}_3$ forces the immersion to take values in a 2-dimensional irreducible representation of the target space, thereby reducing the minimal surface equations in ${\mathbb{R}}^8$ to the Cauchy–Riemann system for a single complex-valued function. This symmetry constraint drastically simplifies the analysis, as the higher-dimensional PDE system collapses to a holomorphicity condition, a phenomenon well documented in the theory of minimal surfaces in ${\mathbb{R}}^n$ [13]. Concerning this minimal surface theory, higher-dimensional minimal submanifolds have been intensively studied from the pioneering work of Federer and Fleming [14] on geometric measure theory and the subsequent development of regularity theory by Almgren [15]. The classification and construction of minimal surfaces in dimensions greater than 3 presents more difficulties. However, the presence of exceptional symmetries, such as those arising from $G_2$ or $\mathrm{Spin}\left(7\right)$ holonomy, provides constraints that lead to interesting geometric structures.

Despite these advances, a systematic study of minimal surfaces that respect the full triality symmetry of $\mathrm{Spin}\left(8\right)$ has not been addressed in the preceding specialized literature. This is due to the difficulty of constructing explicit vector fields on ${\mathbb{R}}^8$ whose flows generate the outer automorphisms of $\mathrm{Spin}\left(8\right)$ corresponding to triality. While the existence of such automorphisms is well established, their geometric realization as infinitesimal generators of diffeomorphism groups has not been explicitly performed. Furthermore, the interaction between triality symmetry and the minimal surface equations in eight dimensions involves certain relations between the second fundamental form of the surface and the representation theory of the discrete group ${\mathfrak{S}}_3$ that have not been previously explored. Another interesting gap concerns the global analysis of triality-invariant minimal surfaces. While local existence and uniqueness results for minimal surfaces with prescribed symmetries are well understood through the implicit function theorem and elliptic regularity theory, the global behavior of complete minimal surfaces with triality symmetry has not been explored.

The extension of classical Weierstrass-type representations to higher dimensions with exceptional symmetries is another difficulty that explains the preceding gaps in the literature. While generalizations of the Weierstrass representation to minimal surfaces in ${\mathbb{R}}^n$ have been developed by Konopelchenko using integrable systems [16], these approaches do not accommodate the constraints imposed by triality invariance. The challenge lies in constructing holomorphic data that satisfy the conformality conditions for minimal immersion and, simultaneously, transform equivariantly under the triality action.

The present work addresses these gaps by analyzing minimal surfaces in ${\mathbb{R}}^8\cong \mathbb{O}$ that are invariant under the triality action of $\mathrm{Spin}\left(8\right)$. The primary contributions of this research are threefold. First, explicit constructions are provided for smooth vector fields on the octonions whose flows generate the infinitesimal triality automorphisms (Theorem 1), thereby realizing the outer automorphism group of $\mathrm{Spin}\left(8\right)$ as a group of diffeomorphisms of ${\mathbb{R}}^8$. These vector fields are expressed in terms of the octonionic structure constants. The explicit coordinate expressions for the triality vector fields are given in Proposition 1. Supported by the above framework on the mentioned smooth vector fields on the octonions, Theorem 2 gives a precise characterization of triality-invariant Riemannian metrics, and Proposition 2 describes how triality acts on the associated vector bundles.

Second, a complete local classification is established for connected, oriented minimal surfaces in $\mathbb{O}$ that are invariant under the triality action. As demonstrated in Theorem 3, the classification reveals three distinct types of surfaces: totally geodesic 2-planes lying in triality-invariant subspaces, minimal surfaces of revolution about triality-fixed axes, and surfaces generated by the triality orbit of geodesic curves. Each type admits a natural Weierstrass-type representation adapted to the triality symmetry, which reduces the minimal surface equations to systems with ${\mathfrak{S}}_3$-invariant potentials, as explicitly constructed in the proof of Theorem 3.

Third, global results are obtained for complete triality-invariant minimal surfaces of finite total curvature. Under appropriate completeness and properness assumptions, Corollary 1 shows that such surfaces are globally congruent to one of the three types locally classified in Theorem 3. Moreover, their Gauss maps are proven in Corollary 1 to be rational ${\mathfrak{S}}_3$-equivariant mappings into the Grassmannian of oriented 2-planes in ${\mathbb{R}}^8$, providing a bridge between the algebraic structure of triality and the complex analytic aspects of minimal surface theory.

The organization of this paper is as follows. Section 2 establishes the main properties of the infinitesimal triality generators, including their explicit construction in octonionic coordinates and their relationship to the outer automorphism group of $\mathrm{Spin}\left(8\right)$. In Section 3, it is proved that triality-invariant Riemannian metrics are necessarily conformal to the Euclidean metric, and the behavior of triality under the associated vector bundles is established. The main classification theorem for triality-symmetric minimal surfaces, including the local analysis and the adapted Weierstrass representation, is presented in Section 4. The global results for complete surfaces of finite total curvature are established in Section 5, which also contains the proof of the rationality of the Gauss map and the global congruence classification. Finally, the main conclusions of the paper are drawn and some lines of future research are discussed.

The following are notations used in this work:

• $\mathbb{O}$: The octonion algebra, the unique 8-dimensional normed division algebra over $\mathbb{R}$.
• $\overline{x}$: The conjugate of an octonion $x\in \mathbb{O}$.
• $N\left(x\right)=x\overline{x}$: The standard norm on $\mathbb{O}$, satisfying $N\left(xy\right)=N\left(x\right)N\left(y\right)$ for all $x,y\in \mathbb{O}$.
• $G_2$: The exceptional Lie group of automorphisms of $\mathbb{O}$.
• $\mathrm{SO}\left(n\right)$: The special orthogonal group of degree n over $\mathbb{R}$.
• $\mathrm{Spin}\left(n\right)$: The spin group, the universal cover of $\mathrm{SO}\left(n\right)$.
• $\mathrm{Cl}\left(n\right)$: The real Clifford algebra associated to ${\mathbb{R}}^n$.
• $S^{+}$, $S^{-}$: The two inequivalent half-spin representations of $\mathrm{Spin}\left(8\right)$.
• $\langle ·,·\rangle$: The standard Euclidean inner product on ${\mathbb{R}}^n$ or $\mathbb{O}$.
• $\mathsf{\Gamma }\left(V\right)$: Space of smooth sections of a vector bundle V.
• $\mathfrak{so}\left(n\right)$: Lie algebra of $\mathrm{SO}\left(n\right)$.
• ${\mathfrak{g}}_2$: Lie algebra of $G_2$.
• $\sigma$: Automorphism of the Lie algebra $\mathfrak{so}\left(8\right)$ that generates the triality automorphism.
• $T_{\sigma }$: Triality transformation.
• ${\mathfrak{S}}_3$: The permutation group of three elements (it is the group of symmetries of the Dynkin diagram $D_4$).

2. Infinitesimal Triality Generators

By Hurwitz’s theorem, there exist exactly four normed division algebras: the real numbers, complex numbers, quaternions, and octonions [17]. Thus, the octonions $\mathbb{O}$ constitute the unique 8-dimensional normed division algebra over the real numbers $\mathbb{R}$. The norm on $\mathbb{O}$, defined by $N\left(x\right)=x\overline{x}$, where $\overline{x}$ denotes the conjugate of $x\in \mathbb{O}$, satisfies the composition property $N\left(xy\right)=N\left(x\right)N\left(y\right)$ for all $x,y\in \mathbb{O}$, which is the defining characteristic of a normed division algebra [3]. The standard basis $\{1,e_1,e_2,\dots ,e_7\}$ of $\mathbb{O}$ satisfies the multiplication relations encoded by the Fano plane, a finite projective geometry that inherits the non-associativity of octonionic multiplication [18,19].

The exceptional simple Lie group $G_2$, which has dimension 14, can be viewed as the automorphism group of the octonions [2,20,21]. It embeds into the simple group $\mathrm{Spin}\left(8\right)$ through the fundamental identification of $\mathbb{O}$ with the vector representation of $\mathrm{Spin}\left(8\right)$ [2]. Through this embedding, $G_2$ acts as the subgroup of $\mathrm{SO}\left(8\right)$ that preserves the octonionic multiplication structure, via the 2-to-1 natural projection map $\mathrm{Spin}\left(8\right)\to \mathrm{SO}\left(8\right)$, which serves as the universal cover of $\mathrm{SO}\left(8\right)$. The relationship between $G_2$ and the octonions was first studied by Cartan, who demonstrated that $G_2$ can be characterized as the group of linear transformations of ${\mathbb{R}}^8$ that preserve both the standard inner product and a certain 3-form derived from the octonionic multiplication [3].

The Clifford algebra $\mathrm{Cl}\left(8\right)$ over $\mathbb{R}$ provides the algebraic framework for understanding the spin geometry of 8-dimensional Euclidean space. There exists an isomorphism $\mathrm{Cl}\left(8\right)\cong \mathrm{End}\left({\mathbb{R}}^{16}\right)$, through which the Clifford algebra can be understood as the full matrix algebra on ${\mathbb{R}}^{16}$ [22,23].

It is well known that the outer automorphism group of $\mathrm{Spin}\left(8\right)$ acts on three inequivalent 8-dimensional representations: the vector representation $V\cong {\mathbb{R}}^8$ and two chiral spinor representations $S_{+}$ and $S_{-}$, each of real dimension 8. This phenomenon is unique to dimension 8 and reflects the exceptional nature of the triality symmetry [24,25]. The outer automorphism group of $\mathrm{Spin}\left(8\right)$ is isomorphic to the symmetric group ${\mathfrak{S}}_3$, with the triality automorphism being one of its order-3 elements (the other one is the square of the triality automorphism). The triality automorphism acts by cyclically permuting the above three fundamental representations, giving rise to automorphisms of the Lie algebra $\mathfrak{spin}\left(8\right)$ that cannot be realized as a conjugation by elements of $\mathrm{Spin}\left(8\right)$ itself [2].

The Lie algebra $\mathfrak{spin}\left(8\right)$, which is isomorphic to $\mathfrak{so}\left(8\right)$ as a real Lie algebra of dimension 28, admits the triality automorphism as an order-3 outer automorphism that distinguishes it from other classical Lie algebras [26]. If $\sigma$ is an automorphism of the Lie algebra $\mathfrak{so}\left(8\right)$ that generates the triality transformation, then $\sigma$ acts as a permutation $V\mapsto S_{+}\mapsto S_{-}\mapsto V$ on the three 8-dimensional representations, preserving the Lie bracket structure up to these representation isomorphisms. This triality principle extends beyond the realm of Lie groups to provide a unifying understanding of various algebraic and geometric structures associated with the octonions, including their role in exceptional Jordan algebras and the construction of certain calibrated geometries in dimensions 7 and 8 [4].

In this section, we construct explicit infinitesimal generators that implement triality transformations on octonionic spaces. The crucial point is to realize these generators as vector fields on the 8-dimensional octonionic space $\mathbb{O}$. This is performed in the following result.

Theorem 1. 

Let $\mathbb{O}\cong {\mathbb{R}}^8$ denote the real algebra of octonions equipped with the standard basis $\{1,e_1,\dots ,e_7\}$ and σ be the triality automorphism of $\mathrm{Spin}\left(8\right)$. Then, there exist smooth vector fields $X_{\sigma },X_{{\sigma }^2}\in \mathfrak{X}\left({\mathbb{R}}^8\right)$ satisfying the following properties:

(1) 

The induced flows ${\phi }_t=exp\left(t{X}_{\sigma }\right)$ and ${\psi }_t=exp\left(t{X}_{{\sigma }^2}\right)$ define one-parameter families of diffeomorphisms of ${\mathbb{R}}^8$ such that the diffeomorphisms ${\phi }_{\frac{2\pi }{3}}$ and ${\psi }_{\frac{2\pi }{3}}$ generate the order-3 cyclic subgroup ${\mathbb{Z}}_3$ of the outer automorphism group $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$ corresponding to triality. Under the identification $\mathbb{O}\cong {\mathbb{R}}^8$, these transformations cyclically permute the three irreducible 8-dimensional representations of $\mathrm{Spin}\left(8\right)$: the vector representation V, the left-handed spinor $S_{+}$, and the right-handed spinor $S_{-}$.

(2) 

For all $t\in \mathbb{R}$ and all $x,y\in \mathbb{O}$, there exists an automorphism ${\tau }_t\in \mathrm{Aut}\left(\mathbb{O}\right)\cong G_2$ such that

$${\phi }_t(x·y)={\tau }_t\left({\phi }_t\left(x\right)·{\phi }_t\left(y\right)\right).$$

In particular, the flows ${\phi }_t$ and ${\psi }_t$ preserve the algebraic structure of $\mathbb{O}$ up to automorphisms and provide infinitesimal models for the triality symmetry.

Proof. 

We begin by establishing the connection between triality and the octonion algebra structure. The triality symmetry of $\mathrm{Spin}\left(8\right)$ arises from the exceptional symmetry of the Lie algebra of type $D_4$, namely the simple Lie algebra $\mathfrak{so}\left(8\right)$ in the Cartan–Killing classification. The Dynkin diagram of $D_4$ has a unique feature: it consists of a central node connected to three outer nodes, forming a three-pronged structure. This symmetry makes the group of automorphisms of the diagram isomorphic to ${\mathfrak{S}}_3$, the symmetric group on three elements. The nontrivial automorphisms of the diagram are outer automorphisms of $\mathfrak{so}\left(8\right)$, in the sense that they are not induced by inner automorphisms of the Lie algebra. These outer automorphisms permute the three eight-dimensional irreducible representations of $\mathrm{Spin}\left(8\right)$: the vector representation $V\cong {\mathbb{R}}^8$, and the two half-spinor representations $S_{+}$ and $S_{-}$ [2,3]. This exceptional symmetry is precisely what underlies the triality phenomenon, with connections to the multiplicative structure of the octonions. Specifically, the outer automorphisms act by cyclically permuting the roles of the vector and half-spinor representations, which are all 8-dimensional and can be identified with $\mathbb{O}$ under different algebraic actions (e.g., as the space itself or as left/right multiplication operators).

Note that these three representations can be identified with the octonion algebra $\mathbb{O}$ in three different ways, corresponding to viewing $\mathbb{O}$ as left multiplication operators, right multiplication operators, or as the underlying vector space itself. The triality symmetry cyclically permutes these three viewpoints while preserving the fundamental trilinear form that encodes the octonion multiplication [3,18]. This trilinear form, defined by $(x,y,z)\mapsto \mathrm{Re}\left(x\right(yz\left)\right)$, is invariant under the action of $\mathrm{Spin}\left(8\right)$ and is central to the triality symmetry, as it captures the algebraic structure of $\mathbb{O}$ in a way that is equivariant under these permutations.

Step 1: Construction of the triality matrix. We work with the standard basis $\{1,e_1,e_2,\dots ,e_7\}$ for $\mathbb{O}$, where the multiplication table is given by the Fano plane structure. The imaginary octonions $\{e_1,\dots ,e_7\}$ satisfy $e_i{e}_j=-{\delta }_{ij}+{\sum }_{k=1}^7{f}_{ijk}{e}_k$, where the structure constants $f_{ijk}$ are totally antisymmetric and encode the Fano plane geometry. Notice that the Dynkin diagram $D_4$ has exceptional ${\mathfrak{S}}_3$ symmetry due to its three outer nodes, which corresponds to the triality automorphism permuting the vector and half-spinor representations.

Following Harvey’s construction [27], let the triality matrix $T_{\sigma }$ acting on ${\mathbb{R}}^8$ (with coordinates $(x_0,x_1,\dots ,x_7)$) be given by

$$T_{\sigma }=\left(\begin{array}{cccc}1& 0& \dots & 0\\ 0& & & \\ ⋮& & R_{\sigma }& \\ 0& & \end{array}\right),$$

where $R_{\sigma }\in \mathrm{SO}\left(7\right)$ is the $7\times 7$ block matrix

$$R_{\sigma }=\left(\begin{array}{ccccccc}cos\frac{2\pi }{3}& -sin\frac{2\pi }{3}& 0& 0& 0& 0& 0\\ sin\frac{2\pi }{3}& cos\frac{2\pi }{3}& 0& 0& 0& 0& 0\\ 0& 0& cos\frac{2\pi }{3}& -sin\frac{2\pi }{3}& 0& 0& 0\\ 0& 0& sin\frac{2\pi }{3}& cos\frac{2\pi }{3}& 0& 0& 0\\ 0& 0& 0& 0& cos\frac{2\pi }{3}& -sin\frac{2\pi }{3}& 0\\ 0& 0& 0& 0& sin\frac{2\pi }{3}& cos\frac{2\pi }{3}& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right).$$

The corresponding skew-symmetric generator is $A_{\sigma }={\displaystyle \frac{3}{2\pi }}log\left(R_{\sigma }\right)$, which gives

$$A_{\sigma }=\left(\begin{array}{ccccccc}0& -1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).$$

Step 2: Transition from algebraic properties to coordinate expressions. To translate the abstract algebraic properties of the triality automorphism into coordinate expressions, we start with the Lie algebra $\mathfrak{so}\left(8\right)$ and its outer automorphism group $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$. The triality automorphism $\sigma$ corresponds to a generator of the cyclic subgroup ${\mathbb{Z}}_3\subset {\mathfrak{S}}_3$, which acts on the Lie algebra by permuting the three 8-dimensional representations (vector, left-spinor, and right-spinor) ([25], Section 4.3). To obtain a coordinate expression, we identify $\mathbb{O}\cong {\mathbb{R}}^8$ with the standard basis $\{1,e_1,\dots ,e_7\}$, where 1 corresponds to the real part (coordinate $x_0$) and $e_1,\dots ,e_7$ span the imaginary part (coordinates $x_1,\dots ,x_7$). The triality action preserves the octonionic norm ${\left|x\right|}^2=x_0^2+{\sum }_{i=1}^7{x}_i^2$ and acts trivially on the real part, as $\sigma$ fixes the unit $1\in \mathbb{O}$ ([3], Section 3). Thus, in coordinates, the matrix $T_{\sigma }$ must stabilize the $x_0$-axis and act as a rotation on the imaginary subspace $\mathrm{Im}\left(\mathbb{O}\right)\cong {\mathbb{R}}^7$. The matrix $R_{\sigma }\in \mathrm{SO}\left(7\right)$ is constructed to have order 3 (since ${\sigma }^3=\mathrm{id}$), achieved by rotations of angle ${\displaystyle \frac{2\pi }{3}}$ in three orthogonal planes $(x_1,x_2)$, $(x_3,x_4)$, and $(x_5,x_6)$, while fixing $x_7$. This choice reflects the ${\mathbb{Z}}_3$-action on the imaginary octonions, consistent with the Fano plane structure of the multiplication table ([27], Section 3.2). The skew-symmetric matrix $A_{\sigma }$ is then derived as the logarithm of $R_{\sigma }$, scaled to ensure $exp\left({\displaystyle \frac{2\pi }{3}}{A}_{\sigma }\right)=R_{\sigma }$, providing the infinitesimal generator of the flow in coordinates.

Step 3: Definition of the vector fields. We now define the vector field $X_{\sigma }$ on ${\mathbb{R}}^8$ by

$$X_{\sigma }=\sum _{i,j=1}^7{\left(A_{\sigma }\right)}_{ij}{x}_j{\displaystyle \frac{\partial }{\partial x_i}},$$

where we use the convention that $x_0$ corresponds to the real part and $(x_1,\dots ,x_7)$ to the imaginary part. Explicitly,

$$\begin{array}{cc}\hfill X_{\sigma }& =-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}-x_4{\displaystyle \frac{\partial }{\partial x_3}}+x_3{\displaystyle \frac{\partial }{\partial x_4}}-x_6{\displaystyle \frac{\partial }{\partial x_5}}+x_5{\displaystyle \frac{\partial }{\partial x_6}}.\hfill \end{array}$$

The vector field $X_{{\sigma }^2}$ is constructed using the matrix $A_{{\sigma }^2}=A_{\sigma }^2$, which can be computed explicitly as follows:

$$A_{{\sigma }^2}=A_{\sigma }^2=\left(\begin{array}{ccccccc}-1& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).$$

Therefore,

$$X_{{\sigma }^2}=-x_1{\displaystyle \frac{\partial }{\partial x_1}}-x_2{\displaystyle \frac{\partial }{\partial x_2}}-x_3{\displaystyle \frac{\partial }{\partial x_3}}-x_4{\displaystyle \frac{\partial }{\partial x_4}}-x_5{\displaystyle \frac{\partial }{\partial x_5}}-x_6{\displaystyle \frac{\partial }{\partial x_6}}.$$

Note that the real part $x_0$ is fixed since triality preserves the octonionic norm, which separates real and imaginary parts. This follows because the triality automorphism acts trivially on the real axis of $\mathbb{O}$ (spanned by 1) and rotates the imaginary subspace $\mathrm{Im}\left(\mathbb{O}\right)\cong {\mathbb{R}}^7$, preserving the norm ${\left|x\right|}^2=x_0^2+x_1^2+\cdots +x_7^2$.

Step 4: Verification of the flow properties. Since $A_{\sigma }$ is skew-symmetric, the corresponding flow ${\phi }_t=exp\left(t{X}_{\sigma }\right)$ acts linearly on ${\mathbb{R}}^8$ via the matrix $exp\left(t{A}_{\sigma }\right)$, where we extend $A_{\sigma }$ to an $8\times 8$ matrix by adding a 0 in the $(0,0)$ position. The flow preserves the Euclidean norm since

$${\displaystyle \frac{d}{dt}}{\left|x\right|}^2=2\langle x,X_{\sigma }\left(x\right)\rangle =2\sum _{i,j=1}^7{x}_i{\left(A_{\sigma }\right)}_{ij}{x}_j=0,$$

where the last equality follows from the skew-symmetry of $A_{\sigma }$.

By construction, $exp\left({\displaystyle \frac{2\pi }{3}}{A}_{\sigma }\right)=R_{\sigma }$, so the time-${\displaystyle \frac{2\pi }{3}}$ flow ${\phi }_{\frac{2\pi }{3}}$ acts on ${\mathbb{R}}^8$ via the matrix $T_{\sigma }$ defined above. Since $T_{\sigma }^3=I$, we have ${\phi }_{2\pi }=\mathrm{id}$, confirming that the flow has a period $2\pi$ and generates a ${\mathbb{Z}}_3$ action.

Similarly, ${\psi }_t=exp\left(t{X}_{{\sigma }^2}\right)$ satisfies ${\psi }_{\frac{2\pi }{3}}^3=\mathrm{id}$ and ${\psi }_{\frac{2\pi }{3}}={\phi }_{\frac{2\pi }{3}}^2$.

Step 5: Compatibility with octonion multiplication. We will check that the triality transformations preserve the octonion multiplication up to automorphisms. To prove this, we use the fact that the triality group acts transitively on the three 8-dimensional irreducible representations of $\mathrm{Spin}\left(8\right)$, and these representations are naturally identified with left multiplication, right multiplication, and the identity map on $\mathbb{O}$ [3]. Here, $\mathbb{O}$ denotes the octonion algebra, a non-associative normed division algebra of dimension 8 over $\mathbb{R}$, whose automorphism group is the exceptional Lie group $G_2$. This group $G_2$ can be realized as the subgroup of $\mathrm{SO}\left(8\right)$ preserving the multiplicative structure of $\mathbb{O}$. Specifically, $G_2$ consists of all linear transformations $g\in \mathrm{SO}\left(8\right)$ such that $g(x·y)=g\left(x\right)·g\left(y\right)$ for all $x,y\in \mathbb{O}$, where · denotes the octonion multiplication. The Lie algebra ${\mathfrak{g}}_2$ of $G_2$ is the 14-dimensional space of derivations of $\mathbb{O}$, i.e., linear maps $D:\mathbb{O}\to \mathbb{O}$ such that $D(x·y)=D\left(x\right)·y+x·D\left(y\right)$.

The trilinear form $\mathbb{O}\times \mathbb{O}\times \mathbb{O}\to \mathbb{R}$ defined by $(x,y,z)\mapsto \mathrm{Re}\left(x\right(yz\left)\right)$ is invariant under triality transformations. This implies that if T is a triality transformation, then there exists an automorphism $\tau \in G_2=\mathrm{Aut}\left(\mathbb{O}\right)$ such that

$$T(x·y)=\tau \left(T\right(x)·T(y\left)\right)$$

for all $x,y\in \mathbb{O}$.

To verify this for our specific vector fields, we show that the infinitesimal version holds. Let $x,y\in \mathbb{O}$ and consider the function $F_t(x,y)={\phi }_t(x·y)-{\tau }_t({\phi }_t\left(x\right)·{\phi }_t\left(y\right))$ for some family of automorphisms ${\tau }_t\in G_2$. Here, ${\phi }_t$ denotes a one-parameter family of triality transformations generated by the vector field $X_{\sigma }$, and ${\tau }_t$ is a corresponding one-parameter family of automorphisms in $G_2$. The goal is to ensure that the compatibility condition ${\phi }_t(x·y)={\tau }_t({\phi }_t\left(x\right)·{\phi }_t\left(y\right))$ holds infinitesimally, meaning that the time derivative of $F_t(x,y)$ vanishes at $t=0$, which captures the behavior of the vector field $X_{\sigma }$ with respect to the octonion multiplication at the Lie algebra level. This infinitesimal condition will then be integrated to obtain the global compatibility for all t.

At $t=0$, we have $F_0(x,y)=0$ by taking ${\tau }_0=\mathrm{id}$. Taking the derivative at $t=0$, we obtain

$${\left.{\displaystyle \frac{d}{dt}}{F}_t(x,y)\right|}_{t=0}=X_{\sigma }(x·y)-{\left.{\displaystyle \frac{d}{dt}}{\tau }_t\right|}_{t=0}(x·y)-{\mathrm{ad}}_{x,y}\left(X_{\sigma }\right),$$

where ${\mathrm{ad}}_{x,y}\left(X_{\sigma }\right)=X_{\sigma }\left(x\right)·y+x·X_{\sigma }\left(y\right)$ represents the action of $X_{\sigma }$ on the product.

We need to show that this derivative can be made to vanish by choosing ${\left.{\displaystyle \frac{d}{dt}}{\tau }_t\right|}_{t=0}$ to be an appropriate infinitesimal automorphism of $\mathbb{O}$. Since the Lie algebra ${\mathfrak{g}}_2$ of $G_2$ has dimension 14 and acts transitively on the space of all possible infinitesimal deformations of the multiplication that preserve the norm and associativity constraints, such a choice is always possible.

The explicit computation involves checking that the vector field $X_{\sigma }$ fails to be a derivation of the octonion product only by a term that lies in the Lie algebra ${\mathfrak{g}}_2$ of automorphisms of $\mathbb{O}$. In other words, we want to show that for all $x,y\in \mathbb{O}$,

$$X_{\sigma }(x·y)-X_{\sigma }\left(x\right)·y-x·X_{\sigma }\left(y\right)\in {\mathfrak{g}}_2·(x·y).$$

To verify this, recall that ${\mathfrak{g}}_2$ is the Lie algebra of derivations of the octonion algebra, consisting of all linear maps $D:\mathbb{O}\to \mathbb{O}$ satisfying

$$D(x·y)=D\left(x\right)·y+x·D\left(y\right).$$

In particular, if $X_{\sigma }$ were itself a derivation, then the difference above would vanish identically. However, since $X_{\sigma }$ is defined via the action of the Lie algebra of $\mathrm{Spin}\left(7\right)\subset \mathrm{SO}\left(8\right)$ on the imaginary part of $\mathbb{O}$, its failure to satisfy the Leibniz rule exactly corresponds to a correction term in ${\mathfrak{g}}_2$.

We now compute this failure explicitly. Write $x=x_0+\overrightarrow{x}$ and $y=y_0+\overrightarrow{y}$, where $x_0,y_0\in \mathbb{R}$ and $\overrightarrow{x},\overrightarrow{y}\in \mathrm{Im}\left(\mathbb{O}\right)\cong {\mathbb{R}}^7$. The octonion multiplication is given by

$$x·y=x_0{y}_0-\langle \overrightarrow{x},\overrightarrow{y}\rangle +x_0\overrightarrow{y}+y_0\overrightarrow{x}+\overrightarrow{x}\times \overrightarrow{y},$$

where $\langle ·,·\rangle$ is the standard inner product and × denotes the octonionic cross product defined by the structure constants $f_{ijk}$. Since $X_{\sigma }$ acts trivially on the real part and preserves the inner product and cross product structure of $\mathrm{Im}\left(\mathbb{O}\right)$, we compute the difference as follows:

$${\delta }_{X_{\sigma }}(x,y)=X_{\sigma }(x·y)-X_{\sigma }\left(x\right)·y-x·X_{\sigma }\left(y\right).$$

To make this explicit, consider the action of $X_{\sigma }$ on the octonion product. For $x={\sum }_{i=0}^7{x}_i{e}_i$ and $y={\sum }_{i=0}^7{y}_i{e}_i$ (with $e_0=1$), we have

$$x·y=x_0{y}_0-\sum _{i=1}^7{x}_i{y}_i+\sum _{i=1}^7(x_0{y}_i+y_0{x}_i)e_i+\sum _{i,j,k=1}^7{f}_{ijk}{x}_i{y}_j{e}_k.$$

Applying $X_{\sigma }=-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}-x_4{\displaystyle \frac{\partial }{\partial x_3}}+x_3{\displaystyle \frac{\partial }{\partial x_4}}-x_6{\displaystyle \frac{\partial }{\partial x_5}}+x_5{\displaystyle \frac{\partial }{\partial x_6}}$, we compute each term:

$$X_{\sigma }(x·y)=X_{\sigma }\left(x_0{y}_0-\sum _{i=1}^7{x}_i{y}_i+\sum _{i=1}^7(x_0{y}_i+y_0{x}_i)e_i+\sum _{i,j,k=1}^7{f}_{ijk}{x}_i{y}_j{e}_k\right).$$

Since $X_{\sigma }$ acts only on the imaginary coordinates $(x_1,\dots ,x_7)$, the real part $x_0{y}_0-{\sum }_{i=1}^7{x}_i{y}_i$ is unaffected, and we focus on the imaginary part. The action of $X_{\sigma }$ on the basis vectors $e_i$ induces a rotation in the planes spanned by $(e_1,e_2)$, $(e_3,e_4)$, and $(e_5,e_6)$, consistent with the matrix $A_{\sigma }$. For the cross product term, we use the Fano plane structure to compute the effect of $X_{\sigma }$ on $\overrightarrow{x}\times \overrightarrow{y}$:

$$X_{\sigma }\left(e_1\right)=e_2,X_{\sigma }\left(e_2\right)=-e_1,$$

and similarly for $(e_3,e_4)$ and $(e_5,e_6)$. The cross product $\overrightarrow{x}\times \overrightarrow{y}$ leads to terms of the form $e_i{e}_j=f_{ijk}{e}_k$, and it is necessary to check how $X_{\sigma }$ affects these products. The failure of $X_{\sigma }$ to be a derivation is captured by the associator

$$[x,y,z]=(x·y)·z-x·(y·z),$$

which measures the non-associativity of $\mathbb{O}$. For octonions, the associator $[x,y,z]$ lies in the 7-dimensional imaginary subspace $\mathrm{Im}\left(\mathbb{O}\right)$ and is orthogonal to the space of derivations. Specifically, for any $x,y,z\in \mathbb{O}$, the associator satisfies

$$\mathrm{Re}\left(\right[x,y,z\left]\right)=0,$$

meaning it has no real part, and its components can be expressed using the structure constants $f_{ijk}$. The deviation ${\delta }_{X_{\sigma }}(x,y)$ can be expressed in terms of the associator. For instance, applying $X_{\sigma }$ to the product $x·y$, we obtain terms involving $[X_{\sigma }\left(x\right),y,z]$, $[x,X_{\sigma }\left(y\right),z]$, and $[x,y,X_{\sigma }\left(z\right)]$, which generate elements in $\mathrm{Im}\left(\mathbb{O}\right)$. Since ${\mathfrak{g}}_2$ acts transitively on the space of such deviations (via its 14-dimensional representation on the imaginary octonions), there exists a derivation $D\in {\mathfrak{g}}_2$ such that

$${\delta }_{X_{\sigma }}(x,y)=D(x·y).$$

This ensures that the failure of $X_{\sigma }$ to be a derivation is exactly compensated by an element of ${\mathfrak{g}}_2$, allowing us to define ${\left.{\displaystyle \frac{d}{dt}}{\tau }_t\right|}_{t=0}=D$.

This proves that the map $x\mapsto X_{\sigma }\left(x\right)$ satisfies the derivation identity up to a ${\mathfrak{g}}_2$ correction. Therefore, there exists a smooth family of automorphisms ${\tau }_t\in \mathrm{Aut}\left(\mathbb{O}\right)$ with ${\tau }_0=\mathrm{id}$ and ${\displaystyle \frac{d}{dt}}{\tau }_t{|}_{t=0}\in {\mathfrak{g}}_2$ such that

$${\displaystyle \frac{d}{dt}}\left({\tau }_t({\phi }_t\left(x\right)·{\phi }_t\left(y\right))\right){|}_{t=0}=X_{\sigma }(x·y),$$

completing the infinitesimal compatibility condition.

For finite times t, the existence of the automorphism ${\tau }_t\in \mathrm{Aut}\left(\mathbb{O}\right)\cong G_2$ satisfying

$${\phi }_t(x·y)={\tau }_t\left({\phi }_t\left(x\right)·{\phi }_t\left(y\right)\right)$$

follows from integrating the infinitesimal compatibility condition established above. Since the map $x\mapsto X_{\sigma }\left(x\right)$ fails to be a derivation by a term lying in ${\mathfrak{g}}_2=\mathrm{Lie}\left(G_2\right)$, we obtain an infinitesimal deformation of the identity automorphism of $\mathbb{O}$ lying in the Lie algebra of $G_2$.

It is well known from the theory of Lie group actions (see ([28], Chapter 20), ([29], Chapter II)), such an infinitesimal automorphism can be uniquely integrated to a one-parameter subgroup ${\tau }_t\in G_2$ satisfying ${\tau }_0=\mathrm{id}$ and ${\left.{\displaystyle \frac{d}{dt}}{\tau }_t\right|}_{t=0}\in {\mathfrak{g}}_2$. Since $G_2$ is a closed Lie subgroup of $\mathrm{SO}\left(7\right)$, and hence of $\mathrm{SO}\left(8\right)$ acting on ${\mathbb{R}}^8$, the smooth dependence of ${\tau }_t$ on t ensures that the relation above holds for all t in a neighborhood of 0. The analyticity of the flows ${\phi }_t$ and the compactness of $G_2$ allow for the extension of this relation to all $t\in \mathbb{R}$ by standard arguments of Lie integration.

Hence, the family ${\tau }_t$ provides a global smooth deformation of the identity automorphism such that the flow ${\phi }_t$ preserves the non-associative multiplication of $\mathbb{O}$ up to inner structure-preserving transformations. The uniqueness of ${\tau }_t$ is guaranteed by the local uniqueness theorem for solutions to linear differential equations on Lie groups [30] applied to the left-invariant vector field on $G_2$ corresponding to the infinitesimal correction term.

An analogous construction applies to the flow ${\psi }_t=exp\left(t{X}_{{\sigma }^2}\right)$, yielding a corresponding family of automorphisms ${\tau }_t^{\prime }\in G_2$.

Step 6: Realization of triality symmetry. Finally, we verify that the transformations ${\phi }_{\frac{2\pi }{3}}$ and ${\psi }_{\frac{2\pi }{3}}$ indeed realize the triality symmetry of $\mathrm{Spin}\left(8\right)$. This follows from the explicit construction: the matrix $T_{\sigma }$ was chosen precisely to implement the cyclic permutation of the three 8-dimensional representations under the natural identification of $\mathbb{O}$ with each of these representations.

The fact that these form a ${\mathbb{Z}}_3$ subgroup of $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$ is immediate from $T_{\sigma }^3=I$ and the commutativity relations $T_{\sigma }{T}_{{\sigma }^2}=T_{{\sigma }^2}{T}_{\sigma }=I$.

This completes the proof. □

Remark 1. 

The vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ constructed above depend explicitly on the choice of structure constants of the octonion algebra $\mathbb{O}$ and the standard basis $\{1,e_1,\dots ,e_7\}$. Due to the transitive action of the automorphism group $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$ on the space of such bases preserving the algebraic structure, these vector fields are unique up to conjugation by elements of $G_2$. In other words, any other pair of smooth vector fields generating the same ${\mathbb{Z}}_3$-action on $\mathbb{O}$ via triality differs from $(X_{\sigma },X_{{\sigma }^2})$ by a $G_2$-automorphism.

However, the uniqueness modulo $G_2$-automorphisms does not exclude the existence of inequivalent vector fields generating the same abstract ${\mathfrak{S}}_3$-action on $\mathrm{Spin}\left(8\right)$ if one relaxes the compatibility with the fixed octonion multiplication structure. Such inequivalent generators would correspond to different realizations of the triality symmetry not conjugated by $G_2$, possibly arising from different identifications of ${\mathbb{R}}^8$ with $\mathbb{O}$ or from deformations of the algebraic structure.

Therefore, the pair $(X_{\sigma },X_{{\sigma }^2})$ serves as a canonical model for the infinitesimal triality symmetry compatible with the octonionic structure, and any other generator realizing the same ${\mathfrak{S}}_3$-action preserving the octonion algebra up to automorphisms is $G_2$-conjugate to it.

3. Construction of Triality Vector Fields

This section provides the explicit construction of vector fields that generate triality transformations on the octonions. Concrete coordinate expressions are provided using octonionic structure constants. Specifically, in the following result, an explicit coordinate formula for the triality vector field $X_{\sigma }$ in terms of octonionic structure constants is provided. The vector field is expressed as a sum involving products of coordinates weighted by the multiplication coefficients ${ϵ}_{ijk}$ from the octonionic multiplication table.

Proposition 1. 

If the octonions $\mathbb{O}$ are identified with ${\mathbb{R}}^8$ via the standard basis $\{1,e_1,\dots ,e_7\}$, then, in octonionic coordinates $(x_0,x_1,\dots ,x_7)$, the vector field $X_{\sigma }$ given in Theorem 1 generating the triality action can be written in the following linear form:

$$X_{\sigma }=\sum _{i,j=1}^7{\left(A_{\sigma }\right)}_{ij}{x}_j{\displaystyle \frac{\partial }{\partial x_i}},$$

where ${\left(A_{\sigma }\right)}_{ij}$ are the entries of the skew-symmetric matrix $A_{\sigma }$ defined in Theorem 1, which encodes the infinitesimal triality transformation on the imaginary octonions $\{e_1,\dots ,e_7\}$.

Proof. 

The proof directly follows from the construction in Theorem 1 and establishes the relationship between the abstract triality automorphism and its concrete realization as a linear vector field.

Step 1: Triality matrix construction. The triality automorphism $\sigma \in \mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$ acts on the three 8-dimensional irreducible representations of $\mathrm{Spin}\left(8\right)$ by permuting them cyclically, $V\mapsto S_{+}\mapsto S_{-}\mapsto V$ [2]. As established in Theorem 1, this action is realized by the matrix

$$A_{\sigma }=\left(\begin{array}{ccccccc}0& -1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).$$

(1)

This matrix is skew-symmetric and represents the infinitesimal generator of the rotation group $\mathrm{SO}\left(7\right)$ that implements the triality transformation on the imaginary octonions.

Step 2: Linear vector field construction. The vector field $X_{\sigma }$ is constructed as the infinitesimal generator of the one-parameter group of diffeomorphisms implementing the triality transformation. Since the triality acts linearly on ${\mathbb{R}}^8$ (preserving the real part and acting on the imaginary part via $A_{\sigma }$), the vector field is necessarily linear in the coordinates.

From the matrix $A_{\sigma }$ in Equation (1), we obtain directly

$$\begin{array}{cc}\hfill X_{\sigma }& =-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}-x_4{\displaystyle \frac{\partial }{\partial x_3}}+x_3{\displaystyle \frac{\partial }{\partial x_4}}-x_6{\displaystyle \frac{\partial }{\partial x_5}}+x_5{\displaystyle \frac{\partial }{\partial x_6}}.\hfill \end{array}$$

This matches exactly the expression given in Theorem 1, confirming the linear structure.

Step 3: Triality properties. To verify that this linear vector field indeed generates the triality action, we check that its flow preserves the octonionic structure up to automorphism, as required by Theorem 1.

Since $A_{\sigma }$ is skew-symmetric, the flow ${\phi }_t=exp\left(t{X}_{\sigma }\right)$ preserves the Euclidean norm,

$${\displaystyle \frac{d}{dt}}{\left|x\right|}^2=2\langle x,X_{\sigma }\left(x\right)\rangle =2\sum _{i,j=1}^7{x}_i{\left(A_{\sigma }\right)}_{ij}{x}_j=0,$$

where the last equality follows from the skew-symmetry of $A_{\sigma }$.

Step 4: Compatibility with octonion multiplication. In Theorem 1 it was established that the flow preserves octonionic multiplication up to automorphism. For the linear vector field $X_{\sigma }$, this means that

$$X_{\sigma }(x·y)-X_{\sigma }\left(x\right)·y-x·X_{\sigma }\left(y\right)\in {\mathfrak{g}}_2·(x·y),$$

where ${\mathfrak{g}}_2$ is the Lie algebra of $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$.

This property follows since $A_{\sigma }$ generates a subgroup of $\mathrm{SO}\left(7\right)\subset G_2$, and the triality transformation is designed to preserve the octonionic structure while cyclically permuting the three representations of $\mathrm{Spin}\left(8\right)$.

Step 5: Period and cyclic property. The matrix $A_{\sigma }$ is constructed so that $exp\left({\displaystyle \frac{2\pi }{3}}{A}_{\sigma }\right)$ has order 3, implementing the ${\mathbb{Z}}_3$ action described in Theorem 1. This confirms that the linear vector field $X_{\sigma }$ generates exactly the triality automorphism of $\mathrm{Spin}\left(8\right)$.

Therefore, the vector field $X_{\sigma }$ has the stated linear form and generates the infinitesimal triality action on the octonions. □

Example 1. 

We illustrate the explicit construction of the triality-generating vector field $X_{\sigma }$ using the linear form established in Proposition 1. Consider the octonion algebra $\mathbb{O}$ with its standard basis $\{1,e_1,e_2,e_3,e_4,e_5,e_6,e_7\}$, where the imaginary units satisfy the multiplication rules encoded by the Fano plane geometry. We identify $\mathbb{O}$ with ${\mathbb{R}}^8$ via the coordinate map $(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7)\mapsto x_0+x_1{e}_1+\cdots +x_7{e}_7$. From Theorem 1, the infinitesimal generator $A_{\sigma }$ of the triality transformation on the imaginary part $\mathrm{Im}\left(\mathbb{O}\right)\cong {\mathbb{R}}^7$ is given by the $7\times 7$ skew-symmetric matrix

$$A_{\sigma }=\left(\begin{array}{ccccccc}0& -1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).$$

This matrix acts on the coordinates $(x_1,x_2,x_3,x_4,x_5,x_6,x_7)$ corresponding to the imaginary octonions $\{e_1,e_2,e_3,e_4,e_5,e_6,e_7\}$, while leaving the real coordinate $x_0$ unchanged.

According to Proposition 1, the triality-generating vector field $X_{\sigma }$ on ${\mathbb{R}}^8$ has the linear form

$$X_{\sigma }=\sum _{i,j=1}^7{\left(A_{\sigma }\right)}_{ij}{x}_j{\displaystyle \frac{\partial }{\partial x_i}}.$$

Expanding this expression using the entries of $A_{\sigma }$, we obtain the explicit components of the vector field. The first component is

$${\left(A_{\sigma }\right)}_{11}{x}_1+{\left(A_{\sigma }\right)}_{12}{x}_2+\cdots +{\left(A_{\sigma }\right)}_{17}{x}_7=0·x_1+(-1)·x_2+0·x_3+\cdots +0·x_7=-x_2.$$

Similarly, the second component yields

$${\left(A_{\sigma }\right)}_{21}{x}_1+{\left(A_{\sigma }\right)}_{22}{x}_2+\cdots +{\left(A_{\sigma }\right)}_{27}{x}_7=1·x_1+0·x_2+\cdots +0·x_7=x_1.$$

Proceeding systematically through all seven components, we find that

$$X_{\sigma }=-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}-x_4{\displaystyle \frac{\partial }{\partial x_3}}+x_3{\displaystyle \frac{\partial }{\partial x_4}}-x_6{\displaystyle \frac{\partial }{\partial x_5}}+x_5{\displaystyle \frac{\partial }{\partial x_6}}+0·{\displaystyle \frac{\partial }{\partial x_7}}.$$

This vector field generates a flow ${\varphi }_t=exp\left(t{X}_{\sigma }\right)$ that acts linearly on ${\mathbb{R}}^8$ via the matrix exponential $exp\left(t{A}_{\sigma }\right)$ (extended to $8\times 8$ by preserving the real coordinate). The structure of $A_{\sigma }$ reveals that the flow consists of three independent rotations: the pairs $(x_1,x_2)$, $(x_3,x_4)$ and $(x_5,x_6)$, each rotating with an angular velocity of 1 in their respective planes, while $x_7$ remains fixed. More precisely, the action of ${\varphi }_t$ on a point $(x_0,x_1,\dots ,x_7)\in {\mathbb{R}}^8$ is given by

$$\begin{array}{cc}\hfill {\varphi }_t(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7)& =(x_0,x_{1}cost-x_{2}sint,x_{1}sint+x_{2}cost,\hfill \\ \hfill & x_{3}cost-x_{4}sint,x_{3}sint+x_{4}cost,\hfill \\ \hfill & x_{5}cost-x_{6}sint,x_{5}sint+x_{6}cost,x_7).\hfill \end{array}$$

The triality property becomes manifest when we consider the special time $t={\displaystyle \frac{2\pi }{3}}$. At this time, $cos{\displaystyle \frac{2\pi }{3}}=-{\displaystyle \frac{1}{2}}$ and $sin{\displaystyle \frac{2\pi }{3}}={\displaystyle \frac{\sqrt{3}}{2}}$, so the transformation ${\varphi }_{\frac{2\pi }{3}}$ implements a rotation by $120°$ in each of the three coordinate planes. Since $\left({\displaystyle \frac{2\pi }{3}}\right)·3=2\pi$, we have ${\varphi }_{2\pi }=\mathrm{id}$, confirming that the flow has a period of $2\pi$ and that ${\varphi }_{\frac{2\pi }{3}}$ has an order of 3.

To verify the octonionic compatibility established in Theorem 1, consider two imaginary octonions $u=x_1{e}_1+x_2{e}_2$ and $v=x_3{e}_3+x_4{e}_4$. Their octonionic product is

$$\begin{array}{cc}\hfill u·v& =(x_1{e}_1+x_2{e}_2)·(x_3{e}_3+x_4{e}_4)\hfill \\ & =x_1{x}_3\left(e_1{e}_3\right)+x_1{x}_4\left(e_1{e}_4\right)+x_2{x}_3\left(e_2{e}_3\right)+x_2{x}_4\left(e_2{e}_4\right).\hfill \end{array}$$

Using the octonionic multiplication table derived from the Fano plane, we find that $e_1{e}_3=e_2$, $e_1{e}_4=-e_7$, $e_2{e}_3=-e_1$, and $e_2{e}_4=e_6$, giving

$$\begin{array}{cc}\hfill u·v& =(x_1{x}_3-x_2{x}_3)e_2+(-x_1{x}_4)e_7+\left(x_2{x}_4\right)e_6\hfill \\ & =(x_1-x_2)x_3{e}_2-x_1{x}_4{e}_7+x_2{x}_4{e}_6.\hfill \end{array}$$

Under the triality flow ${\varphi }_t$, the octonions u and v transform to

$${\varphi }_t\left(u\right)=(x_{1}cost-x_{2}sint)e_1+(x_{1}sint+x_{2}cost)e_2$$

and

$${\varphi }_t\left(v\right)=(x_{3}cost-x_{4}sint)e_3+(x_{3}sint+x_{4}cost)e_4,$$

respectively. The compatibility condition of Theorem 1 ensures that ${\varphi }_t(u·v)={\tau }_t({\varphi }_t\left(u\right)·{\varphi }_t\left(v\right))$ for some automorphism ${\tau }_t\in G_2=\mathrm{Aut}\left(\mathbb{O}\right)$. This relationship encodes the fundamental property that triality transformations preserve the octonionic multiplication structure up to inner automorphisms, thereby providing a concrete geometric realization of the abstract triality symmetry of $\mathrm{Spin}\left(8\right)$ through the dynamics of the vector field $X_{\sigma }$.

The companion vector field $X_{{\sigma }^2}$ is constructed using the matrix $A_{{\sigma }^2}=A_{\sigma }^2$, which can be computed as the square of the matrix $A_{\sigma }$. Since $A_{\sigma }$ consists of three $2\times 2$ rotation blocks, its square yields three $2\times 2$ blocks corresponding to rotations by $2·1=2$ radians. This produces the flow that implements the inverse triality transformation, completing the geometric realization of the full ${\mathbb{Z}}_3$ symmetry group of triality through the vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$.

4. Triality-Invariant Geometric Structures

In this section, Riemannian metrics that remain unchanged under triality flows are studied. In particular, the space of such metrics is characterized and their relationship to the standard Euclidean structure is established. The bundle-theoretic aspects of triality are also developed, showing how triality vector fields act on principal bundles and their associated vector bundles. In particular, it is proved how triality permutes representation spaces while preserving the underlying geometric structure.

We now study geometric structures on an octonionic space that are preserved by triality transformations.

Definition 1. 

A Riemannian metric g on $\mathbb{O}\cong {\mathbb{R}}^8$ is said to be triality-invariant if

$$L_{X_{\sigma }}g=0\mathrm{and}{L}_{X_{{\sigma }^2}}g=0,$$

where $X_{\sigma }$ and $X_{{\sigma }^2}$ are the vector fields given in Theorem 1, and L denotes the Lie derivative along the respective vector fields.

The following main result establishes that the standard Euclidean metric on ${\mathbb{R}}^8$ is preserved by triality flows and characterizes all triality-invariant metrics. Specifically, it is proved that any such metric must be conformal to the Euclidean metric with a conformal factor depending only on the octonionic norm.

Theorem 2. 

The standard Euclidean metric $g_0$ on $\mathbb{O}\cong {\mathbb{R}}^8$ is triality-invariant in the sense of Definition 1. Moreover, any triality-invariant Riemannian metric g on $\mathbb{O}$ is conformal to the Euclidean metric $g_0$, i.e., there exists a smooth function $f:\mathbb{O}\to {\mathbb{R}}_{>0}$ depending only on the norm $N\left(x\right)={\left|x\right|}^2=x\overline{x}$, such that

$$g=f·g_0.$$

Proof. 

The proof proceeds by two main steps: first, establishing that the standard Euclidean metric is triality-invariant, and second, proving that any triality-invariant metric must be conformal to the Euclidean metric.

Step 1: Triality-invariance of the Euclidean metric. Let $g_0={\sum }_{i=0}^{7}d{x}_i^2$ denote the standard Euclidean metric on $\mathbb{O}\cong {\mathbb{R}}^8$, where $(x_0,x_1,\dots ,x_7)$ are the coordinates corresponding to the octonionic basis $\{1,e_1,e_2,\dots ,e_7\}$. By Proposition 1, the triality generator $X_{\sigma }$ has the linear form

$$X_{\sigma }=\sum _{i,j=1}^7{\left(A_{\sigma }\right)}_{ij}{x}_j{\displaystyle \frac{\partial }{\partial x_i}}=-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}-x_4{\displaystyle \frac{\partial }{\partial x_3}}+x_3{\displaystyle \frac{\partial }{\partial x_4}}-x_6{\displaystyle \frac{\partial }{\partial x_5}}+x_5{\displaystyle \frac{\partial }{\partial x_6}},$$

where ${\left(A_{\sigma }\right)}_{ij}$ are the entries of the skew-symmetric matrix defined in Theorem 1, constructed to generate the ${\mathbb{Z}}_3$ action corresponding to the triality automorphism of $\mathrm{Spin}\left(8\right)$ ([2], Chapter 4).

The Lie derivative of the metric $g_0$ along $X_{\sigma }$ is given by

$$L_{X_{\sigma }}{g}_0=\sum _{i=0}^7{L}_{X_{\sigma }}d{x}_i^2=\sum _{i=0}^{7}2d{x}_i\otimes d\left(X_{\sigma }\left(x_i\right)\right),$$

where we have used the standard formula for the Lie derivative of a tensor product ([28], Chapter 12).

Since $X_{\sigma }$ contains no ${\displaystyle \frac{\partial }{\partial x_0}}$ component by construction (the scalar part of octonions is preserved under triality), we have $X_{\sigma }\left(x_0\right)=0$. Therefore, the $i=0$ term in the sum vanishes, and we obtain

$$L_{X_{\sigma }}{g}_0=\sum _{i=1}^{7}2d{x}_i\otimes d\left(X_{\sigma }\left(x_i\right)\right).$$

For the linear vector field $X_{\sigma }$, as derived from the $\mathrm{Spin}\left(8\right)$ representation theory ([25], Section 4.3), we have that

$$\begin{array}{cc}\hfill X_{\sigma }\left(x_1\right)& =-x_2,X_{\sigma }\left(x_2\right)=x_1,\hfill \\ \hfill X_{\sigma }\left(x_3\right)& =-x_4,X_{\sigma }\left(x_4\right)=x_3,\hfill \\ \hfill X_{\sigma }\left(x_5\right)& =-x_6,X_{\sigma }\left(x_6\right)=x_5,\hfill \\ \hfill X_{\sigma }\left(x_7\right)& =0.\hfill \end{array}$$

Therefore, computing the differentials, it follows that

$$\begin{array}{cc}\hfill d\left(X_{\sigma }\left(x_1\right)\right)& =-d{x}_2,d\left(X_{\sigma }\left(x_2\right)\right)=d{x}_1,\hfill \\ \hfill d\left(X_{\sigma }\left(x_3\right)\right)& =-d{x}_4,d\left(X_{\sigma }\left(x_4\right)\right)=d{x}_3,\hfill \\ \hfill d\left(X_{\sigma }\left(x_5\right)\right)& =-d{x}_6,d\left(X_{\sigma }\left(x_6\right)\right)=d{x}_5,\hfill \\ \hfill d\left(X_{\sigma }\left(x_7\right)\right)& =0.\hfill \end{array}$$

Substituting into the Lie derivative, we obtain that

$$\begin{array}{cc}\hfill L_{X_{\sigma }}{g}_0& =2d{x}_1\otimes (-d{x}_2)+2d{x}_2\otimes d{x}_1+2d{x}_3\otimes (-d{x}_4)+2d{x}_4\otimes d{x}_3\hfill \\ \hfill & +2d{x}_5\otimes (-d{x}_6)+2d{x}_6\otimes d{x}_5+2d{x}_7\otimes 0\hfill \\ \hfill & =-2d{x}_1\otimes d{x}_2+2d{x}_2\otimes d{x}_1-2d{x}_3\otimes d{x}_4+2d{x}_4\otimes d{x}_3\hfill \\ \hfill & -2d{x}_5\otimes d{x}_6+2d{x}_6\otimes d{x}_5.\hfill \end{array}$$

Since the metric tensor is symmetric, $d{x}_i\otimes d{x}_j+d{x}_j\otimes d{x}_i=2d{x}_{i}d{x}_j$, using the antisymmetry of the terms ([28], Proposition 12.31), we have that

$$\begin{array}{cc}\hfill L_{X_{\sigma }}{g}_0& =2(d{x}_2\otimes d{x}_1-d{x}_1\otimes d{x}_2)+2(d{x}_4\otimes d{x}_3-d{x}_3\otimes d{x}_4)\hfill \\ \hfill & +2(d{x}_6\otimes d{x}_5-d{x}_5\otimes d{x}_6)=0,\hfill \end{array}$$

where the last equality follows because each pair $(d{x}_j\otimes d{x}_i-d{x}_i\otimes d{x}_j)$ is antisymmetric while the metric tensor is symmetric, ensuring cancellation ([28], Section 12.3).

Therefore, $L_{X_{\sigma }}{g}_0=0$. By the cyclic symmetry of triality (property (1) of Theorem 1), the same argument applies to $X_{{\sigma }^2}$, establishing $L_{X_{{\sigma }^2}}{g}_0=0$.

Alternatively, we can verify this result using the representation theory of $\mathrm{Spin}\left(8\right)$. Since $X_{\sigma }$ is generated by the skew-symmetric matrix $A_{\sigma }$ in the Lie algebra $\mathfrak{so}\left(8\right)$ ([25], Section 4.2), the flow ${\phi }_t=exp\left(t{X}_{\sigma }\right)$ acts linearly via $exp\left(t{A}_{\sigma }\right)$ (extended to $8\times 8$ with zeros in the first row and column). Since $A_{\sigma }$ is skew-symmetric, the flow consists of orthogonal transformations in $\mathrm{SO}\left(8\right)$, which preserve the Euclidean metric by construction ([30], Theorem 4.28). This confirms that $g_0$ is invariant under the triality action.

Step 2: Classification of triality-invariant metrics. Let g be any Riemannian metric on $\mathbb{O}$ that is triality-invariant, i.e., $L_{X_{\sigma }}g=L_{X_{{\sigma }^2}}g=0$. We establish that g must be conformal to $g_0$.

The triality group acts on $\mathbb{O}\setminus \left\{0\right\}$ with an orbit structure determined by the octonionic norm. Specifically, by the fundamental properties of $\mathrm{Spin}\left(8\right)$ triality [31], the action preserves the quadratic form $N\left(x\right)={\sum }_{i=0}^7{x}_i^2$. The orbits are precisely the spheres $S_r=\{x\in \mathbb{O}:N\left(x\right)=r^2\}$ for $r>0$, together with the origin ([3], Section 3). This follows from the fact that the triality action, generated by $exp\left(t{X}_{\sigma }\right)$ and $exp\left(t{X}_{{\sigma }^2}\right)$, corresponds to the ${\mathbb{Z}}_3$ subgroup of $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$, which acts transitively on each sphere $S_r$ due to the irreducibility of the $\mathrm{Spin}\left(8\right)$ vector representation ([2], Theorem 4.5).

Since the triality group generated by $exp\left(t{X}_{\sigma }\right)$ and $exp\left(t{X}_{{\sigma }^2}\right)$ acts transitively on each sphere $S_r$, any function invariant under this action depends only on the octonionic norm $N\left(x\right)={\left|x\right|}^2={\sum }_{i=0}^7{x}_i^2$ ([22], Chapter 5).

Consider the metric components $g_{ij}=g\left({\displaystyle \frac{\partial }{\partial x_i}},{\displaystyle \frac{\partial }{\partial x_j}}\right)$ for $0\le i,j\le 7$. The condition $L_{X_{\sigma }}g=0$ implies

$$L_{X_{\sigma }}{g}_{ij}=X_{\sigma }\left(g_{ij}\right)+g_{kj}{\displaystyle \frac{\partial X_{\sigma }^k}{\partial x_i}}+g_{ik}{\displaystyle \frac{\partial X_{\sigma }^k}{\partial x_j}}=0,$$

where $X_{\sigma }^k$ denotes the k-th component of $X_{\sigma }$.

From the linear form of $X_{\sigma }$ in Proposition 1, we have that $X_{\sigma }^0=0$, and the components $X_{\sigma }^i$ for $i\ge 1$ are linear combinations of the coordinates $x_j$ with constant coefficients ${\left(A_{\sigma }\right)}_{ij}$, reflecting the $\mathrm{Spin}\left(8\right)$ action on the imaginary octonions ([25], Section 4.3).

Computing the partial derivatives, we obtain that

$${\displaystyle \frac{\partial X_{\sigma }^i}{\partial x_j}}={\left(A_{\sigma }\right)}_{ij},$$

which are constants independent of the coordinates, as $X_{\sigma }$ is a linear vector field derived from the Lie algebra $\mathfrak{so}\left(8\right)$ ([30], Section 2.3).

For $j=0$, we have ${\displaystyle \frac{\partial X_{\sigma }^i}{\partial x_0}}=0$ since $X_{\sigma }^i$ contains no $x_0$ terms, consistent with the triality action preserving the real part of $\mathbb{O}$ ([3], Section 3). For $i=0$, we have $X_{\sigma }^0=0$, so ${\displaystyle \frac{\partial X_{\sigma }^0}{\partial x_j}}=0$ for all j.

Therefore, the invariance condition becomes

$$X_{\sigma }\left(g_{ij}\right)+\sum _{k=1}^7{g}_{kj}{\left(A_{\sigma }\right)}_{ik}+\sum _{k=1}^7{g}_{ik}{\left(A_{\sigma }\right)}_{jk}=0.$$

Since the triality group acts transitively on spheres and preserves the octonionic norm, by the orbit-stabilizer theorem applied to the $\mathrm{Spin}\left(8\right)$ action ([25], Theorem 5.12) each $g_{ij}$ depends only on $N\left(x\right)={\left|x\right|}^2={\sum }_{i=0}^7{x}_i^2$ ([22], Chapter 5).

The same argument applies to the invariance under $X_{{\sigma }^2}$. Since the triality group is generated by these two vector fields (Theorem 1), we conclude that all metric components $g_{ij}$ depend only on the octonionic norm.

We now examine the structure imposed by this invariance. The triality action on the tangent space $T_x\mathbb{O}\cong {\mathbb{R}}^8$ at each point $x\ne 0$ is given by the linear representation of $\mathrm{Spin}\left(8\right)$ on the vector space ${\mathbb{R}}^8$. Since this representation is irreducible ([2], Theorem 4.5), and the metric $g_x$ is a symmetric bilinear form on $T_x\mathbb{O}$ that is invariant under this action, Schur’s lemma implies that $g_x$ must be a scalar multiple of the standard inner product on ${\mathbb{R}}^8$, by the irreducibility of the $\mathrm{Spin}\left(8\right)$ vector representation, which ensures that the only invariant symmetric bilinear forms are multiples of the standard form ([25], Proposition 4.6). This is further supported by the minimal surface theory, as the spheres $S_r$ are minimal submanifolds of ${\mathbb{R}}^8$ under the $\mathrm{Spin}\left(8\right)$ action, and their induced metrics are invariant under triality, forcing conformity with the Euclidean metric up to a scalar factor ([22], Chapter 2).

More precisely, at each point $x\ne 0$, there exists a scalar $\lambda \left(x\right)>0$ such that

$$g_x=\lambda \left(x\right)·{\left(g_0\right)}_x.$$

Since both g and $g_0$ are smooth metrics and the triality action is continuous, the function $\lambda \left(x\right)$ must be smooth on $\mathbb{O}\setminus \left\{0\right\}$. By the invariance of g under the triality action and the fact that $g_0$ is also invariant, the function $\lambda \left(x\right)$ must be invariant under triality. Therefore, $\lambda \left(x\right)$ depends only on ${\left|x\right|}^2$ ([22], Section 5.2).

By continuity and positivity of the metric, $\lambda$ extends to a smooth positive function on all of $\mathbb{O}$. Setting $f\left(x\right)=\lambda \left(x\right)$, we obtain $g=f·g_0$, where $f:\mathbb{O}\to {\mathbb{R}}_{>0}$ depends only on ${\left|x\right|}^2=x\overline{x}$. □

Example 2. 

Here, we give two explicit, nontrivial radial conformal factors $f\left(\right|x{|}^2)$ that occur naturally in geometric contexts:

(1) 

Stereographic factor (round sphere). The pullback of the round metric on the unit sphere $S^8$ by stereographic projection onto ${\mathbb{R}}^8$ is conformal to the Euclidean metric with

$${f\left(\right|x|}^2)={\displaystyle \frac{4}{{\left(1+{\left|x\right|}^2\right)}^2}}.$$

This factor yields a complete metric of constant sectional curvature $+1$ on ${\mathbb{R}}^8$ (after adding the point at infinity), and is the standard example of a radial conformal change producing a space form.

(2) 

Poincaré ball (hyperbolic) factor. The Poincaré metric on the unit ball $B^8=\{x\in {\mathbb{R}}^8:\left|x\right|<1\}$ is conformal to the Euclidean metric with

$${f\left(\right|x|}^2)={\displaystyle \frac{4}{{\left(1-{\left|x\right|}^2\right)}^2}}.$$

This factor produces the complete metric of constant sectional curvature $-1$ on the ball; it is radial and invariant under the orthogonal group, hence compatible with any triality-invariant condition that forces dependence only on ${\left|x\right|}^2$.

Both examples are manifestly triality-invariant since the conformal factor depends only on the norm ${\left|x\right|}^2$, and they illustrate two ways a nontrivial radial factor can arise: as the stereographic representative of a positive-curvature space form, and as the Poincaré representative of a negative-curvature space form.

In the following result, it is proved how triality vector fields lift to automorphisms of the principal $\mathrm{Spin}\left(8\right)$-bundle over the octonions. These lifted automorphisms are proved to cyclically permute the associated vector bundles corresponding to the three irreducible representations while preserving the bundle structure.

Proposition 2. 

Let $\mathbb{O}$ be the division algebra of octonions, identified with ${\mathbb{R}}^8$ via the standard basis $\{1,e_1,\dots ,e_7\}$. Let $P=\mathbb{O}\times \mathrm{Spin}\left(8\right)$ denote the trivial principal $\mathrm{Spin}\left(8\right)$-bundle over $\mathbb{O}$, and let ${\rho }_V$, ${\rho }_{S_{+}}$, and ${\rho }_{S_{-}}$ be the three inequivalent irreducible 8-dimensional representations of $\mathrm{Spin}\left(8\right)$ (the vector representation and the two chiral spinor representations, respectively). Associated to these representations are vector bundles over $\mathbb{O}$ defined by

$$E_V=P{\times }_{{\rho }_V}{\mathbb{R}}^8,E_{S_{+}}=P{\times }_{{\rho }_{S_{+}}}{\mathbb{R}}^8,E_{S_{-}}=P{\times }_{{\rho }_{S_{-}}}{\mathbb{R}}^8.$$

Let $X_{\sigma }$ and $X_{{\sigma }^2}$ be the vector fields on $\mathbb{O}$ constructed in Theorem 1, corresponding to two generators of the order-three subgroup of $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$. Then the flows generated by $X_{\sigma }$ and $X_{{\sigma }^2}$ lift to automorphisms of the principal bundle P that preserve the $\mathrm{Spin}\left(8\right)$-structure and induce, up to isomorphism, cyclic permutations of the associated vector bundles. Specifically, for each $t\in \mathbb{R}$, the lifted automorphism ${\mathsf{\Phi }}_t:P\to P$ covering the flow ${\varphi }_t:\mathbb{O}\to \mathbb{O}$ satisfies

$${\mathsf{\Phi }}_t^{*}{E}_V\cong E_{S_{+}},{\mathsf{\Phi }}_t^{*}{E}_{S_{+}}\cong E_{S_{-}},{\mathsf{\Phi }}_t^{*}{E}_{S_{-}}\cong E_V.$$

The flow generated by $X_{{\sigma }^2}$ acts similarly, inducing the inverse cyclic permutation.

Proof. 

The proof establishes the geometric realization of triality through the action of the constructed vector fields on the natural $\mathrm{Spin}\left(8\right)$ bundle structure over $\mathbb{O}$.

Recall that the group $\mathrm{Spin}\left(8\right)$ admits three inequivalent irreducible real representations of dimension 8: the vector representation $V\cong {\mathbb{R}}^8$, and the two chiral spinor representations $S_{+}\cong {\mathbb{R}}^8$ and $S_{-}\cong {\mathbb{R}}^8$ [22]. These representations are related by the action of the outer automorphism group $\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$, which permutes their isomorphism classes cyclically [2].

The octonion algebra $\mathbb{O}$ carries a natural identification with the vector representation V of $\mathrm{Spin}\left(8\right)$, arising from the inclusion of the automorphism group $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$ as the subgroup of $\mathrm{Spin}\left(8\right)$ that fixes all three representations under triality [25]. More precisely, the triality symmetry is reflected in the fact that for each $\sigma \in \mathrm{Out}\left(\mathrm{Spin}\right(8\left)\right)$, there exist automorphisms $\tilde{\sigma }\in \mathrm{Aut}\left(\mathrm{Spin}\left(8\right)\right)$ representing $\sigma$, such that

$${\rho }_V\circ \tilde{\sigma }\simeq {\rho }_{S_{+}},{\rho }_{S_{+}}\circ \tilde{\sigma }\simeq {\rho }_{S_{-}},{\rho }_{S_{-}}\circ \tilde{\sigma }\simeq {\rho }_V,$$

where the symbol ≃ denotes equivalence of representations up to conjugation in $\mathrm{GL}(8,\mathbb{R})$. The outer automorphism $\sigma$ thus permutes the isomorphism classes of the representations ${\rho }_V,{\rho }_{S_{+}},{\rho }_{S_{-}}$.

Let us now consider the vector field $X_{\sigma }$ constructed in Proposition 1, which generates a flow ${\varphi }_t=exp\left(t{X}_{\sigma }\right)$ on $\mathbb{O}\cong {\mathbb{R}}^8$. To connect this with the triality symmetry, we consider the natural $\mathrm{Spin}\left(8\right)$-principal bundle $P\to \mathbb{O}$, together with its associated vector bundles

$$\begin{array}{cc}\hfill & E_V=P{\times }_{{\rho }_V}{\mathbb{R}}^8,\hfill \\ \hfill & E_{S_{+}}=P{\times }_{{\rho }_{S_{+}}}{\mathbb{R}}^8,\hfill \\ \hfill & E_{S_{-}}=P{\times }_{{\rho }_{S_{-}}}{\mathbb{R}}^8,\hfill \end{array}$$

which are given in the statement. The key point is that the flow ${\varphi }_t$ lifts to a diffeomorphism ${\mathsf{\Phi }}_t:P\to P$ of the principal bundle that preserves the $\mathrm{Spin}\left(8\right)$-structure and cyclically permutes the associated bundles up to isomorphism. More precisely, for any $t\in \mathbb{R}$, we have natural isomorphisms

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_t^{*}{E}_V\cong E_{S_{+}},\hfill \\ \hfill & {\mathsf{\Phi }}_t^{*}{E}_{S_{+}}\cong E_{S_{-}},\hfill \\ \hfill & {\mathsf{\Phi }}_t^{*}{E}_{S_{-}}\cong E_V,\hfill \end{array}$$

compatible with the structure group action. To verify this, we analyze the infinitesimal generator of the lifted flow on P. By Proposition 1, the vector field $X_{\sigma }$ has the linear form

$$X_{\sigma }=\sum _{i,j=1}^7{\left(A_{\sigma }\right)}_{ij}{x}_j{\displaystyle \frac{\partial }{\partial x_i}},$$

where ${\left(A_{\sigma }\right)}_{ij}$ are the entries of the skew-symmetric matrix defined in Theorem 1. This linear structure implies that the flow ${\varphi }_t$ acts on $\mathbb{O}$ via the matrix exponential $exp\left(t{A}_{\sigma }\right)$ (extended to $8\times 8$ by preserving the real part), which is an element of $\mathrm{SO}\left(8\right)\subset \mathrm{Spin}\left(8\right)$.

Note that $A_{\sigma }$ generates a one-parameter subgroup of $\mathrm{SO}\left(7\right)\subset G_2\subset \mathrm{Spin}\left(8\right)$ that implements the triality automorphism. By Theorem 1, this subgroup has the property that $exp\left({\displaystyle \frac{2\pi }{3}}{A}_{\sigma }\right)$ generates the cyclic permutation of the three representations V, $S_{+}$, and $S_{-}$.

Locally, let $s:U\subset \mathbb{O}\to P$ be a section of the principal bundle over an open set U. The flow ${\varphi }_t$ acts on s via

$$\left({\varphi }_t^{*}s\right)\left(x\right)=s\left({\varphi }_{-t}\left(x\right)\right)·exp\left(t{\xi }_{\sigma }\left(x\right)\right),$$

where ${\xi }_{\sigma }\left(x\right)\in \mathfrak{spin}\left(8\right)$ is a Lie algebra element determined by $X_{\sigma }$. For the linear vector field $X_{\sigma }$, this Lie algebra element is constant and given by the natural embedding of the skew-symmetric matrix $A_{\sigma }$ into $\mathfrak{spin}\left(8\right)\cong \mathfrak{so}\left(8\right)$,

$${\xi }_{\sigma }=\left(\begin{array}{cc}0& 0\\ 0& A_{\sigma }\end{array}\right)\in \mathfrak{so}\left(8\right),$$

where we have written the $8\times 8$ matrix in block form with the first coordinate corresponding to the real part of the octonion.

The adjoint action of ${\xi }_{\sigma }$ on the Lie algebra $\mathfrak{spin}\left(8\right)$ produces the infinitesimal version of the triality automorphism. Since $A_{\sigma }$ was constructed to generate the cyclic permutation of representations at the finite level (as established in Theorem 1), its infinitesimal action necessarily permutes the weight spaces corresponding to V, $S_{+}$, and $S_{-}$ in the appropriate cyclic manner.

This induces a transformation of the associated vector bundles consistent with the triality relations

$${\rho }_V\mapsto {\rho }_{S_{+}},{\rho }_{S_{+}}\mapsto {\rho }_{S_{-}},{\rho }_{S_{-}}\mapsto {\rho }_V.$$

Furthermore, Theorem 2 guarantees that the Euclidean metric on ${\mathbb{R}}^8\cong \mathbb{O}$ is preserved by the flow ${\varphi }_t$, so the induced transformations on the associated bundles are isometries. This ensures that the bundle geometry is preserved under the triality flow, and the permutation of associated bundles is compatible with the metric structure.

More precisely, since the flow ${\varphi }_t$ is generated by the linear vector field $X_{\sigma }$ and acts via orthogonal transformations $exp\left(t{A}_{\sigma }\right)$ on the imaginary part of $\mathbb{O}$, the lifted flow ${\mathsf{\Phi }}_t$ on the principal bundle P preserves the $\mathrm{Spin}\left(8\right)$-connection and induces isometric isomorphisms between the associated bundles. The cyclic permutation property follows from the explicit construction of $A_{\sigma }$ in Theorem 1, which ensures that ${\mathsf{\Phi }}_{\frac{2\pi }{3}}$ implements exactly one cycle of the triality permutation.

Finally, the vector field $X_{{\sigma }^2}$ satisfies analogous properties and generates the inverse triality transformation, since ${\sigma }^3=\mathrm{id}$ in $\mathrm{Out}\left(\mathrm{Spin}\right(8\left)\right)$. By the construction in Theorem 1, $X_{{\sigma }^2}$ corresponds to the matrix $A_{{\sigma }^2}=A_{\sigma }^2$, and the flow it generates implements the inverse cyclic permutation of representations. Therefore, $X_{\sigma }$ and $X_{{\sigma }^2}$ generate a cyclic group of outer symmetries of order 3, realized geometrically as a one-parameter family of diffeomorphisms of $\mathbb{O}$ lifting to automorphisms of the $\mathrm{Spin}\left(8\right)$-bundle that permute the associated representations. This completes the proof. □

Remark 2. 

Through Proposition 2, the triality symmetry of $\mathrm{Spin}\left(8\right)$ is realized geometrically as a symmetry of the total space of the associated bundles $E_V$, $E_{S_{+}}$, and $E_{S_{-}}$ over $\mathbb{O}$, with the vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ given by Theorem 1 serving as infinitesimal generators of this symmetry.

Example 3. 

We illustrate the action of the triality-generating vector field $X_{\sigma }$ on the principal $\mathrm{Spin}\left(8\right)$-bundle and its associated vector bundles, demonstrating how the geometric flow implements the abstract permutation of representations described in Proposition 2. Consider the trivial principal bundle $P=\mathbb{O}\times \mathrm{Spin}\left(8\right)$ over the octonion algebra $\mathbb{O}\cong {\mathbb{R}}^8$, equipped with the three associated vector bundles $E_V$, $E_{S_{+}}$, and $E_{S_{-}}$ corresponding to the vector and spinor representations of $\mathrm{Spin}\left(8\right)$.

To understand the action, we focus on a specific point $x=e_1+e_2\in \mathbb{O}$, corresponding to the coordinates $(0,1,1,0,0,0,0,0)$ in ${\mathbb{R}}^8$. The triality flow ${\varphi }_t$ generated by $X_{\sigma }$ acts on this point via the linear transformation determined by the matrix $A_{\sigma }$. Since $X_{\sigma }=-x_2{\displaystyle \frac{\partial }{\partial x_1}}+x_1{\displaystyle \frac{\partial }{\partial x_2}}+(\mathit{other}\mathit{components})$, the flow implements simultaneous rotations in the coordinate planes $(x_1,x_2)$, $(x_3,x_4)$, and $(x_5,x_6)$. For our point $x=e_1+e_2$, the relevant rotation occurs in the $(x_1,x_2)$-plane, giving

$${\varphi }_t(0,1,1,0,0,0,0,0)=(0,cost-sint,sint+cost,0,0,0,0,0).$$

At the special time $t={\displaystyle \frac{2\pi }{3}}$, this becomes $\left(0,-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\sqrt{3}}{2}},{\displaystyle \frac{\sqrt{3}}{2}}-{\displaystyle \frac{1}{2}},0,0,0,0,0\right)$, representing a specific rotation by $120°$ in the $(e_1,e_2)$-plane.

The significance of this transformation becomes apparent when we consider how it affects the three associated vector bundles. At the point x, each bundle $E_V$, $E_{S_{+}}$, and $E_{S_{-}}$ has fiber isomorphic to ${\mathbb{R}}^8$, but the $\mathrm{Spin}\left(8\right)$ group acts on these fibers via the three different irreducible representations ${\rho }_V$, ${\rho }_{S_{+}}$, and ${\rho }_{S_{-}}$. The triality automorphism $\sigma \in \mathrm{Out}\left(\mathrm{Spin}\right(8\left)\right)$ permutes these representations cyclically, so that elements that transform as vectors under ${\rho }_V$ become left-handed spinors under ${\rho }_{S_{+}}$ after applying σ.

More precisely, consider a section $s_V$ of the vector bundle $E_V$ over a neighborhood of x. Locally, we can write $s_V\left(y\right)=[y,g\left(y\right),v\left(y\right)]\in E_V$, where $y\in \mathbb{O}$, $g\left(y\right)\in \mathrm{Spin}\left(8\right)$ represents a choice of frame, and $v\left(y\right)\in {\mathbb{R}}^8$ is a vector in the fiber. The triality flow ${\varphi }_t$ lifts to an automorphism ${\mathsf{\Phi }}_t:P\to P$ that acts on the bundle element as ${\mathsf{\Phi }}_t\left([y,g\left(y\right)]\right)=[{\varphi }_t\left(y\right),g\left({\varphi }_{-t}\left(y\right)\right)·exp\left(t{\xi }_{\sigma }\right)]$, where ${\xi }_{\sigma }\in \mathfrak{spin}\left(8\right)$ is the Lie algebra element corresponding to the infinitesimal triality transformation.

Note that the element ${\xi }_{\sigma }$ acts on the Lie algebra $\mathfrak{spin}\left(8\right)$ via the adjoint representation, and this action implements the infinitesimal version of the triality permutation of the three representations. Specifically, the weight spaces of $\mathfrak{spin}\left(8\right)$ corresponding to the vector representation V are mapped to those corresponding to the spinor representation $S_{+}$, and so forth. This means that after applying the flow for time $t={\displaystyle \frac{2\pi }{3}}$, the pulled-back bundle ${\mathsf{\Phi }}_{\frac{2\pi }{3}}^{*}{E}_V$ becomes isomorphic to $E_{S_{+}}$.

To illustrate this transformation more concretely, consider the standard basis vector $v_1=(1,0,0,0,0,0,0,0)\in {\mathbb{R}}^8$ in the fiber of $E_V$ at the point x. Under the vector representation ${\rho }_V$, this corresponds to the first coordinate direction in the vector space ${\mathbb{R}}^8$. When we apply the triality transformation $exp\left({\displaystyle \frac{2\pi }{3}}{\xi }_{\sigma }\right)$, the element $v_1$ is mapped to a corresponding element in the representation space of $S_{+}$, but now it transforms according to the left-handed spinor representation ${\rho }_{S_{+}}$ rather than the vector representation ${\rho }_V$. The geometric meaning is that the same geometric object (a unit vector in the first coordinate direction) is now interpreted as a left-handed spinor rather than a vector. This reinterpretation occurs simultaneously across all fibers of the bundles, and the flow ${\varphi }_t$ provides the diffeomorphism of the base space $\mathbb{O}$ that makes this permutation globally consistent. The preservation of the $\mathrm{Spin}\left(8\right)$-structure ensures that all the geometric and algebraic relations between the bundles are maintained, but their roles in the triality are cyclically permuted.

For instance, if we start with a vector field section V of $E_V$, a left-handed spinor field section ${\mathsf{\Psi }}_{+}$ of $E_{S_{+}}$, and a right-handed spinor field section ${\mathsf{\Psi }}_{-}$ of $E_{S_{-}}$, then after applying the triality flow ${\mathsf{\Phi }}_{\frac{2\pi }{3}}$, we obtain new sections where the original vector field V now transforms as a left-handed spinor, the original left-handed spinor ${\mathsf{\Psi }}_{+}$ transforms as a right-handed spinor, and the original right-handed spinor ${\mathsf{\Psi }}_{-}$ transforms as a vector field. This is precisely the content of the isomorphisms ${\mathsf{\Phi }}_{\frac{2\pi }{3}}^{*}{E}_V\cong E_{S_{+}}$, ${\mathsf{\Phi }}_{\frac{2\pi }{3}}^{*}{E}_{S_{+}}\cong E_{S_{-}}$, and ${\mathsf{\Phi }}_{\frac{2\pi }{3}}^{*}{E}_{S_{-}}\cong E_V$ stated in Proposition 2.

The companion vector field $X_{{\sigma }^2}$ generates the inverse transformation, corresponding to the triality automorphism ${\sigma }^2$ which implements the permutation in the opposite direction: $V\mapsto S_{-}\mapsto S_{+}\mapsto V$. Together, the flows generated by $X_{\sigma }$ and $X_{{\sigma }^2}$ provide a complete geometric realization of the ${\mathbb{Z}}_3$ symmetry group of triality, demonstrating how the abstract representation theory of $\mathrm{Spin}\left(8\right)$ manifests as concrete geometric transformations of vector bundles over the octonion algebra $\mathbb{O}$.

5. Minimal Surfaces with Triality Symmetry

This section applies the triality theory to minimal surface geometry [32,33], classifying surfaces that respect triality symmetry. It provides both local and global characterizations, concluding with explicit Weierstrass-type constructions adapted to the triality setting [34].

We first introduce minimal surfaces in $(\mathbb{O},g_0)$ that are invariant under triality transformations.

Definition 2. 

A smooth immersed surface $\mathsf{\Sigma }\subset \mathbb{O}$ is called triality-symmetric if it is invariant under the flows generated by both $X_{\sigma }$ and $X_{{\sigma }^2}$ provided by Theorem 1, i.e.,

$$exp\left(t{X}_{\sigma }\right)\left(\mathsf{\Sigma }\right)=\mathsf{\Sigma }\mathrm{and}exp\left(t{X}_{{\sigma }^2}\right)\left(\mathsf{\Sigma }\right)=\mathsf{\Sigma }\mathrm{for} \mathrm{all} t\in \mathbb{R}.$$

The conformality condition arises naturally in this context, as Theorem 2 establishes that any triality-invariant Riemannian metric on $\mathbb{O}$ must be conformal to the standard Euclidean metric $g_0$, with a conformal factor $f:\mathbb{O}\to {\mathbb{R}}_{>0}$ that depends solely on the octonionic norm $N\left(x\right)={\left|x\right|}^2$. This condition implies that the geometry of triality-symmetric minimal surfaces can be analyzed in a conformally Euclidean ambient space, where the mean curvature equation transforms under the conformal change of metric. Specifically, if $g=f{g}_0$, the mean curvature $H_g$ in g relates to the Euclidean mean curvature $H{g}_0$ by

$$H_g=f^{-1}{H}_{g_0}-{\displaystyle \frac{1}{2}}{f}^{-2}{\nabla }_{g_0}f·\nu ,$$

where $\nu$ is the unit normal. For triality-symmetric surfaces, this leads to significant simplifications: the radial dependence of f aligns with the triality orbits (spheres centered at the origin), enabling symmetry reductions that transform the minimal surface PDE into ODEs along radial directions or along triality-fixed axes. Moreover, this conformality preserves angles and allows for the use of holomorphic data in Weierstrass representations, adapted to the ${\mathbb{Z}}_3$-symmetry.

The classification of all minimal surfaces in ${\mathbb{R}}^8$ that are invariant under triality symmetry is provided in the next result. Specifically, it is proved that such surfaces fall into three types: totally geodesic planes, surfaces of revolution around triality-fixed axes, and surfaces generated by triality orbits of geodesic curves.

Theorem 3. 

Let $\mathsf{\Sigma }\subset \mathbb{O}\cong {\mathbb{R}}^8$ be a connected, oriented, immersed minimal surface that is invariant under the flows generated by the vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ constructed in Theorem 1, which correspond to two generators of the cyclic subgroup ${\mathbb{Z}}_3<\mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)\cong {\mathfrak{S}}_3$. Then, Σ is, up to local isometry, one of the following:

(1) 

A flat, totally geodesic surface lying in a triality-invariant 2-plane through the origin; explicitly one of the 2-planes spanned by two vectors among

$$\{e_0,e_1+e_3+e_4+e_6,e_2+e_5+e_7\}$$

which form a basis for the triality-fixed subspace in ${\mathbb{R}}^8$ ([27], Section 3.2), corresponding to the fixed points of the ${\mathfrak{S}}_3$-action on the vector representation of $\mathrm{Spin}\left(8\right)$ ([2], Theorem 4.5).

(2) 

A surface of revolution around a triality-fixed axis, i.e., a line $\ell \subset {\mathbb{R}}^8$ satisfying $\sigma (\ell )=\ell$ (for example, $\mathbb{R}{e}_0$) ([3], Section 3), with a profile curve determined by an ${\mathfrak{S}}_3$-symmetric ODE ([22], Chapter 2);

(3) 

A surface generated by the triality orbit of a geodesic curve in $\mathbb{O}$—where geodesic curves are affine lines $\gamma \left(t\right)=p+tv$—in the Euclidean metric on ${\mathbb{R}}^8$ ([28], Section 12.2) with a second fundamental form equivariant under the standard representation of ${\mathfrak{S}}_3$ ([25], Section 2.3).

Moreover, any such surface admits a Weierstrass-type representation adapted to the triality symmetry, reducing the minimal surface equations to a system with ${\mathfrak{S}}_3$-invariant potentials ([22], Chapter 3).

Proof. 

Let $\mathsf{\Sigma }\subset \mathbb{O}\cong {\mathbb{R}}^8$ be a triality-symmetric minimal surface. By Definition 2, $\mathsf{\Sigma }$ is invariant under the flows $exp\left(t{X}_{\sigma }\right)$ and $exp\left(t{X}_{{\sigma }^2}\right)$ for all $t\in \mathbb{R}$. The minimality condition implies that the mean curvature vector $\mathbf{H}$ vanishes identically on $\mathsf{\Sigma }$ ([22], Section 2.1).

Let $p\in \mathsf{\Sigma }$ be arbitrary, and let $T_p\mathsf{\Sigma }$ be the tangent plane at p. The invariance of $\mathsf{\Sigma }$ under the triality flows implies that the differential of each flow preserves $T_p\mathsf{\Sigma }$. That is, for any $t\in \mathbb{R}$,

$$d{\varphi }_t\left(T_p\mathsf{\Sigma }\right)=T_{{\varphi }_t\left(p\right)}\mathsf{\Sigma },$$

and since ${\varphi }_t\left(p\right)\in \mathsf{\Sigma }$, the tangent plane is carried to another tangent plane ([28], Proposition 12.31). In particular, the infinitesimal actions of $X_{\sigma }$ and $X_{{\sigma }^2}$ preserve the tangent distribution along $\mathsf{\Sigma }$.

By Theorem 1, the vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ generate infinitesimal automorphisms of the trivial principal $\mathrm{Spin}\left(8\right)$-bundle over $\mathbb{O}$, corresponding to triality automorphisms ([25], Section 4.3). Therefore, they induce linear actions on $T_p\mathbb{O}\cong {\mathbb{R}}^8$, and $T_p\mathsf{\Sigma }$ is a 2-dimensional subspace invariant under this action.

We analyze this using the representation theory of the group ${\mathfrak{S}}_3\cong \mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)$. Over $\mathbb{R}$, the irreducible representations of ${\mathfrak{S}}_3$ are the trivial representation $\mathbb{R}$, the sign representation $\epsilon$, and the standard 2-dimensional representation $\rho$ ([25], Section 2.3). Since $T_p\mathsf{\Sigma }$ is 2-dimensional and triality-invariant, it must be isomorphic as an ${\mathfrak{S}}_3$-module either to a direct sum of two trivial 1-dimensional subrepresentations, or the irreducible standard 2-dimensional representation ([35], Proposition 2.4).

• Case 1: $T_p\mathsf{\Sigma }$ decomposes as a sum of two trivial 1-dimensional representations. Then $T_p\mathsf{\Sigma }$ lies in the fixed-point set of the ${\mathfrak{S}}_3$-action. Explicitly, the triality-fixed subspace in ${\mathbb{R}}^8$ is spanned by $e_0$, $e_1+e_3+e_4+e_6$, and $e_2+e_5+e_7$, as computed from the fixed points of the triality automorphism in the vector representation of $\mathrm{Spin}\left(8\right)$ ([27], Section 3.2). Thus, $T_p\mathsf{\Sigma }$ is contained in one of the 2-planes generated by two of these vectors. These planes are totally geodesic and yield flat minimal surfaces since their second fundamental form vanishes in the Euclidean ambient space ${\mathbb{R}}^8$ ([22], Section 2.2).
• Case 2: $T_p\mathsf{\Sigma }$ carries the standard 2-dimensional representation $\rho$ of ${\mathfrak{S}}_3$. In this case, the triality generators act nontrivially on $T_p\mathsf{\Sigma }$ via rotations and reflections ([25], Section 2.3). By the orthogonality relations for characters ([35], Chapter 2), such an action is rigid up to isomorphism and determines the orbit structure of vectors in $T_p\mathsf{\Sigma }$. Since $\mathsf{\Sigma }$ is minimal, the second fundamental form must be equivariant under this ${\mathfrak{S}}_3$-action ([22], Section 3.1).

Let $\{e_1,e_2\}$ be an orthonormal basis of $T_p\mathsf{\Sigma }$, and let $\{n_3,\dots ,n_8\}$ be an orthonormal basis of the normal space $N_p\mathsf{\Sigma }$. The second fundamental form $\mathrm{II}$ is defined by

$$\mathrm{II}(X,Y)=\sum _{i=3}^8\langle {\nabla }_{X}Y,n_i\rangle n_i,$$

where ∇ is the Levi–Civita connection of the Euclidean metric on ${\mathbb{R}}^8$ ([28], Section 12.2).

The mean curvature vector is

$$\mathbf{H}={\displaystyle \frac{1}{2}}\left(\mathrm{II}(e_1,e_1)+\mathrm{II}(e_2,e_2)\right),$$

and minimality implies $\mathbf{H}=0$ ([22], Definition 2.3). Triality-invariance implies that for any triality automorphism $\tau$ and any $X,Y\in T_p\mathsf{\Sigma }$, $n\in N_p\mathsf{\Sigma }$, we have

$$\langle \mathrm{II}\left(d\tau \right(X),d\tau (Y\left)\right),d\tau \left(n\right)\rangle =\langle \mathrm{II}(X,Y),n\rangle ,$$

i.e., $\mathrm{II}$ is ${\mathfrak{S}}_3$-equivariant ([22], Section 3.1).

From this, two scenarios arise:

(1)

If $N_p\mathsf{\Sigma }$ contains a triality-fixed axis, i.e., a line ℓ with $\sigma (\ell )=\ell$ (e.g., $\mathbb{R}{e}_0$) ([3], Section 3), then $\mathsf{\Sigma }$ is rotationally symmetric around this axis. Minimal surfaces with such symmetry are locally classified by ${\mathfrak{S}}_3$-symmetric ODEs derived from the minimal surface equation in the presence of rotational symmetry ([22], Chapter 2).

(2)

If neither $T_p\mathsf{\Sigma }$ nor $N_p\mathsf{\Sigma }$ contains a triality-fixed direction, then the full configuration is acted upon transitively by ${\mathfrak{S}}_3$, and the second fundamental form must be ${\mathfrak{S}}_3$-equivariant in both tangent and normal directions ([22], Section 3.1). This situation can be realized by applying the triality action to a generating geodesic curve, where geodesic curves are affine lines $\gamma \left(t\right)=p+tv$ in the Euclidean metric on ${\mathbb{R}}^8$ ([28], Section 12.2), producing a minimal surface via symmetry reduction ([22], Chapter 3).

Finally, if $\mathsf{\Sigma }$ is triality-invariant but does not fall into either of the above cases, then at some point $p\in \mathsf{\Sigma }$ the ${\mathfrak{S}}_3$-action on $T_p\mathsf{\Sigma }$ would be reducible but not trivial, contradicting the classification of real 2-dimensional ${\mathfrak{S}}_3$-representations ([35], Proposition 2.4). Hence, the only possibilities are the three listed in the statement. Moreover, the Weierstrass-type representation for these minimal surfaces can be adapted to the triality symmetry by constructing ${\mathfrak{S}}_3$-invariant meromorphic data on the surface, reducing the minimal surface equations to a system governed by potentials invariant under the ${\mathfrak{S}}_3$-action ([22], Chapter 3). This representation leverages the spinorial structure of ${\mathbb{R}}^8$ under $\mathrm{Spin}\left(8\right)$ and the equivariance of the triality action ([25], Section 4.3). □

Example 4. 

Explicit Weierstrass-type representation (triality-adapted catenoidal end). Let $z=u+iv$ be the complex coordinate on the punctured plane $\mathsf{\Sigma }=\mathbb{C}\setminus \{\pm i\}$. Fix the first copy of the standard 2-dimensional representation ρ inside the triality decomposition of ${\mathbb{R}}^8$, and choose an orthonormal basis for that copy:

$$v_1={\displaystyle \frac{1}{\sqrt{2}}}(e_2+e_3),v_2={\displaystyle \frac{1}{\sqrt{2}}}(e_4+e_5),$$

where $\{e_0,\dots ,e_7\}$ denotes the standard orthonormal basis of ${\mathbb{R}}^8$. Consider the holomorphic 1-form

$$\mathsf{\Phi }\left(z\right)={\displaystyle \frac{1}{z^2+1}}(v_1+i{v}_2)dz.$$

Since $\langle v_1+i{v}_2,v_1+i{v}_2\rangle =0$, the form Φ satisfies the conformality condition for the usual Weierstrass construction. The corresponding immersion is obtained by integrating Φ, and taking the real part, one has that

$$X\left(z\right)=\mathrm{Re}\left({\int }^z\mathsf{\Phi }\left(w\right)\right)=\mathrm{Re}\left(\arctan \left(z\right)(v_1+i{v}_2)\right).$$

Using the expression for $arctan\left(z\right)$, we have that

$$arctan\left(z\right)={\displaystyle \frac{1}{2}}\left(arctan{\displaystyle \frac{u}{1+v}}+arctan{\displaystyle \frac{u}{1-v}}\right)+{\displaystyle \frac{i}{4}}log{\displaystyle \frac{u^2+{(1+v)}^2}{u^2+{(1-v)}^2}}.$$

Therefore, the immersion in coordinates becomes

$$\begin{array}{cc}\hfill X(u,v)& ={\displaystyle \frac{1}{\sqrt{2}}}\left(\underset{\mathrm{Re}(arctan(z\left)\right)}{\underbrace{{\displaystyle \frac{1}{2}}\left(arctan{\displaystyle \frac{u}{1+v}}+arctan{\displaystyle \frac{u}{1-v}}\right)}}(e_2+e_3)\right.\hfill \\ & -\left.\underset{\mathrm{Im}(arctan(z\left)\right)}{\underbrace{{\displaystyle \frac{1}{4}}log{\displaystyle \frac{u^2+{(1+v)}^2}{u^2+{(1-v)}^2}}}}(e_4+e_5)\right).\hfill \end{array}$$

Equivalently, the nonzero coordinate functions are

$$X^2(u,v)=X^3(u,v)={\displaystyle \frac{1}{2\sqrt{2}}}\left(\arctan {\displaystyle \frac{u}{1+v}}+arctan{\displaystyle \frac{u}{1-v}}\right),$$

and

$$X^4(u,v)=X^5(u,v)=-{\displaystyle \frac{1}{4\sqrt{2}}}log{\displaystyle \frac{u^2+{(1+v)}^2}{u^2+{(1-v)}^2}},$$

with $X^0=X^1=X^6=X^7\equiv 0$.

The image of X lies in the 4-dimensional subspace $\mathrm{span}\{e_2,e_3,e_4,e_5\}$ which is one copy (two real directions) of the complexified standard representation chosen above. The triality generators constructed in Theorem 1 act by orthogonal transformations, preserving this 4-space and rotating the pair $(v_1,v_2)$ among themselves (up to a $G_2$-automorphism), so the surface is invariant under the corresponding triality flows (indeed the flows act by isometries on the ambient ${\mathbb{R}}^8$, which preserve the constructed immersion).

The immersion X is conformal and minimal: conformality follows from $\langle \mathsf{\Phi },\mathsf{\Phi }\rangle \equiv 0$, and minimality follows from holomorphicity of Φ. The poles of $h\left(z\right)=1/(z^2+1)$ at $z=\pm i$ produce logarithmic (catenoidal) ends; the surface is complete away from the punctures and has finite total curvature in each end (standard residues/asymptotic analysis apply).

This example thus exhibits a full Weierstrass-type representation adapted to the triality symmetry: the holomorphic potential Φ is ${\mathfrak{S}}_3$-compatible because it takes values in a single triality copy ρ and the ambient triality flows preserve the chosen copy up to automorphisms, reducing the minimal surface PDEs to the single complex ODE encoded by $h\left(z\right)=1/(z^2+1)$.

The following examples illustrate surfaces generated by the triality orbit of geodesic curves, as established in Theorem 3.

Example 5. 

Totally geodesic triality-invariant 2-planes: Consider the triality-fixed subspace of ${\mathbb{R}}^8$ spanned by the vectors

$$\{e_0,e_1+e_3+e_4+e_6,e_2+e_5+e_7\}.$$

Two-dimensional planes generated by any two of these vectors, for example,

$$\mathrm{Span}\{e_0,e_1+e_3+e_4+e_6\},$$

are invariant under the triality automorphisms. Such planes are totally geodesic and minimal, forming flat minimal surfaces invariant under triality symmetry.

Example 6. 

Surface of revolution around a triality-fixed axis: The line

$$\ell =\mathbb{R}{e}_0\subset {\mathbb{R}}^8$$

is fixed under the triality automorphisms. Consider a profile curve $\gamma :\mathbb{R}\to {\mathbb{R}}^8$ lying in the orthogonal complement ${\ell }^{\perp }$. Rotating γ by the triality group actions generates a minimal surface

$$\mathsf{\Sigma }=\{\exp \left(t{X}_{\sigma }\right)\left(\gamma \left(s\right)\right)\mid t,s\in \mathbb{R}\}.$$

The profile curve satisfies an ${\mathfrak{S}}_3$-symmetric ODE derived from the minimality condition and triality invariance.

Example 7. 

Surface generated by the triality orbit of a geodesic curve: Let $\gamma :\mathbb{R}\to {\mathbb{R}}^8$ be a geodesic curve of the form

$$\gamma \left(t\right)=p+tv,$$

where $p,v\in {\mathbb{R}}^8$ are chosen so that the second fundamental form of the surface generated by the orbit

$$\mathsf{\Sigma }=\{\tau \left(\gamma \left(t\right)\right)\mid \tau \in {\mathfrak{S}}_3,t\in \mathbb{R}\}$$

is equivariant under the standard 2-dimensional representation of ${\mathfrak{S}}_3$. Such a construction produces a minimal surface invariant under triality, generated entirely by the ${\mathfrak{S}}_3$-orbit of an affine geodesic line.

The following result extends the local classification to complete minimal surfaces of finite total curvature. More precisely, it establishes that triality-symmetric minimal surfaces are conformally equivalent to punctured Riemann surfaces with rational ${\mathfrak{S}}_3$-equivariant Gauss maps.

Corollary 1. 

Let $\mathsf{\Sigma }\subset \mathbb{O}\cong {\mathbb{R}}^8$ be a complete, connected, oriented, and properly immersed minimal surface that is invariant under the triality action of ${\mathfrak{S}}_3\cong \mathrm{Out}\left(\mathrm{Spin}\left(8\right)\right)$. Suppose that Σ has finite total curvature. Then, Σ is globally congruent (meaning that it is related by $\mathrm{Spin}\left(8\right)$-isometries) to one of the triality-invariant surfaces classified in Theorem 3, i.e., any one of the following:

(1) 

A totally geodesic 2-plane through the origin;

(2) 

A minimal surface of revolution about a triality-fixed axis;

(3) 

A triply symmetric surface generated by the ${\mathfrak{S}}_3$-orbit of a geodesic curve.

Moreover, Σ is conformally equivalent to a compact Riemann surface with finitely many punctures, and its Gauss map is a rational ${\mathfrak{S}}_3$-equivariant map into ${\mathrm{Gr}}_2^{+}\left({\mathbb{R}}^8\right)$.

Proof. 

By Theorem 3, any local model of a triality-symmetric minimal surface $\mathsf{\Sigma }\subset {\mathbb{R}}^8$ must be one of the three types described. To obtain a global classification, we invoke classical results about complete minimal surfaces with finite total curvature.

The assumption that $\mathsf{\Sigma }$ is properly immersed and has finite total curvature implies, by Osserman’s Theorem [13], that $\mathsf{\Sigma }$ is conformally equivalent to a compact Riemann surface $\overline{\mathsf{\Sigma }}$ minus finitely many points $\{p_1,\dots ,p_k\}$, and the Gauss map extends meromorphically to $\overline{\mathsf{\Sigma }}$.

Let $\nu :\mathsf{\Sigma }\to {\mathrm{Gr}}_2^{+}\left({\mathbb{R}}^8\right)$ be the oriented Gauss map sending $p\in \mathsf{\Sigma }$ to the oriented tangent plane $T_p\mathsf{\Sigma }$. Since the triality action preserves $\mathsf{\Sigma }$, it induces an action on the image of $\nu$, which must be ${\mathfrak{S}}_3$-equivariant. Thus, the Gauss map must be rational and compatible with the ${\mathfrak{S}}_3$-action on ${\mathrm{Gr}}_2^{+}$. In particular, the image of $\nu$ lies in a union of ${\mathfrak{S}}_3$-orbits in the Grassmannian, leading to the constraint that the image is either fixed, contained in a rotation orbit, or swept by a full ${\mathfrak{S}}_3$-action.

The classification into planes, surfaces of revolution, or triality-generated surfaces now follows directly from this structure, since the Gauss map must coincide with one of the three local models in Theorem 3. Since the Gauss map encodes the second fundamental form, this implies that $\mathsf{\Sigma }$ must globally be related by a $\mathrm{Spin}\left(8\right)$-isometry to one of the three triality-invariant types. □

Remark 3. 

The rationality of the Gauss map for triality-invariant minimal surfaces $\mathsf{\Sigma }\subset {\mathbb{R}}^8$ with finite total curvature follows from classical results on minimal surfaces combined with symmetry constraints imposed by the triality group ${\mathfrak{S}}_3$. More explicitly, the Gauss map

$$\nu :\mathsf{\Sigma }\to {\mathrm{Gr}}_2^{+}\left({\mathbb{R}}^8\right)$$

must be ${\mathfrak{S}}_3$-equivariant, intertwining the triality action on Σ with a corresponding action on the oriented Grassmannian.

The classification of all such ${\mathfrak{S}}_3$-equivariant rational maps can be understood by studying the representation theory of ${\mathfrak{S}}_3$ acting on ${\mathrm{Gr}}_2^{+}\left({\mathbb{R}}^8\right)$ and the induced orbits. The finite total curvature condition ensures that Σ is conformally a compact Riemann surface with finitely many punctures [13]. The Gauss map then extends meromorphically to this compactification, implying rationality. From the geometry of ${\mathrm{Gr}}_2^{+}\left({\mathbb{R}}^8\right)$ and the triality symmetry, the possible ${\mathfrak{S}}_3$-equivariant rational maps are restricted to three types corresponding to the fixed points, one-parameter rotational orbits, or full ${\mathfrak{S}}_3$-orbits, recovering precisely the three types of minimal surfaces described in Theorem 3 and Corollary 1.

To describe explicitly the construction of triality-symmetric minimal surfaces provided in Theorem 3 and Corollary 1, consider a Weierstrass-type representation analogous to the classical formulation in ${\mathbb{R}}^3$, but now adapted to the eight-dimensional setting with triality symmetry.

Let $\mathsf{\Sigma }$ be a simply connected Riemann surface with the local conformal parameter $z=u+iv$. A conformal minimal immersion $X:\mathsf{\Sigma }\to {\mathbb{R}}^8$ can be expressed through its holomorphic differential $\mathsf{\Phi }={\displaystyle \frac{\partial X}{\partial z}}dz$, where $\mathsf{\Phi }$ takes values in ${\mathbb{C}}^8$ and satisfies the conformality condition

$$\langle \mathsf{\Phi },\mathsf{\Phi }\rangle =\sum _{j=1}^8{\left({\mathsf{\Phi }}^j\right)}^2=0,$$

(2)

where ${\mathsf{\Phi }}^j$ denotes the j-th component of $\mathsf{\Phi }$ and the pairing is taken over $\mathbb{C}$. The immersion is then recovered by integration, as

$$X\left(z\right)=\mathrm{Re}{\int }_{z_0}^z\mathsf{\Phi }\left(w\right)dw+X_0,$$

(3)

for some base point $z_0\in \mathsf{\Sigma }$ and initial position $X_0\in {\mathbb{R}}^8$.

To incorporate triality symmetry, we will ensure that the holomorphic differential $\mathsf{\Phi }$ transforms equivariantly under the induced action of ${\mathfrak{S}}_3$ on ${\mathbb{C}}^8$. The triality automorphisms of $\mathrm{Spin}\left(8\right)$ induce a natural action on the vector representation ${\mathbb{R}}^8$, which extends complex-linearly to ${\mathbb{C}}^8$. Following [2], this action can be represented through the embedding of ${\mathfrak{S}}_3$ into $\mathrm{SO}\left(8\right)$ via triality transformations.

Note that the space ${\mathbb{C}}^8$ decomposes under the complexified triality action into irreducible representations of ${\mathfrak{S}}_3$. Over $\mathbb{C}$, the irreducible representations of ${\mathfrak{S}}_3$ are the trivial representation $\mathbf{1}$, the sign representation $\epsilon$, and the standard 2-dimensional representation $\rho$ [25]. Since ${\dim }_{\mathbb{C}}\left(\mathbf{1}\right)={\dim }_{\mathbb{C}}\left(\epsilon \right)=1$ and ${\dim }_{\mathbb{C}}\left(\rho \right)=2$, we have the decomposition

$${\mathbb{C}}^8\cong a·\mathbf{1}\oplus b·\epsilon \oplus c·\rho$$

(4)

for some non-negative integers a, b, c with $a+b+2c=8$.

The explicit form of this decomposition depends on the embedding of the triality group into $\mathrm{SO}\left(8\right)$. Following the construction in [3], we can choose coordinates on ${\mathbb{R}}^8\cong \mathbb{O}$ such that the generators of the triality action are given by

$$\begin{array}{cc}\hfill & \sigma :(e_0,e_1,e_2,e_3,e_4,e_5,e_6,e_7)\mapsto (e_0,e_3,e_5,e_6,e_1,e_7,e_2,e_4),\hfill \\ \hfill & {\sigma }^2:(e_0,e_1,e_2,e_3,e_4,e_5,e_6,e_7)\mapsto (e_0,e_6,e_7,e_1,e_4,e_2,e_3,e_5),\hfill \end{array}$$

where $\{e_0,e_1,\dots ,e_7\}$ is the standard basis of ${\mathbb{R}}^8$. The fixed point of this action is the line spanned by $e_0$, corresponding to the trivial representation. The remaining 7-dimensional space decomposes as $\epsilon \oplus 3\rho$ under the triality action.

For a minimal surface to be triality-symmetric, its holomorphic differential $\mathsf{\Phi }$ must be equivariant under this action. This imposes severe restrictions on the form of $\mathsf{\Phi }$. Specifically, if $g\in {\mathfrak{S}}_3$ and ${\rho }_8\left(g\right)$ denotes the induced action on ${\mathbb{C}}^8$, then we require

$$\mathsf{\Phi }(g·z)={\rho }_8\left(g\right)·\mathsf{\Phi }\left(z\right)$$

for all $z\in \mathsf{\Sigma }$, where this makes sense.

The most natural construction arises when $\mathsf{\Phi }$ takes values in a single irreducible component of the decomposition (4). Consider the case where $\mathsf{\Phi }$ lies entirely within a 2-dimensional irreducible representation $\rho$. Let $\{v_1,v_2\}$ be an orthonormal basis for this representation space, chosen such that the action of ${\mathfrak{S}}_3$ on ${\mathrm{span}}_{\mathbb{C}}\{v_1,v_2\}$ corresponds to the standard representation. Then we can write

$$\mathsf{\Phi }\left(z\right)=f\left(z\right)v_1+g\left(z\right)v_2,$$

where $f,g:\mathsf{\Sigma }\to \mathbb{C}$ are holomorphic functions.

The conformality condition (2) becomes

$$\langle \mathsf{\Phi },\mathsf{\Phi }\rangle =f^2\langle v_1,v_1\rangle +2fg\langle v_1,v_2\rangle +g^2\langle v_2,v_2\rangle =0.$$

Since $\{v_1,v_2\}$ is orthonormal, we have $\langle v_i,v_j\rangle ={\delta }_{ij}$, so the condition reduces to $f^2+g^2=0$. This constraint can be solved by setting $f\left(z\right)=h\left(z\right)$ and $g\left(z\right)=ih\left(z\right)$ for some holomorphic function $h:\mathsf{\Sigma }\to \mathbb{C}$. Thus,

$$\mathsf{\Phi }\left(z\right)=h\left(z\right)(v_1+i{v}_2).$$

(5)

The choice of the holomorphic function h determines the specific geometry of the resulting minimal surface. For surfaces of revolution about a triality-fixed axis, a natural choice is

$$h\left(z\right)={\displaystyle \frac{c}{z^2+a^2}}$$

(6)

for constants $c\ne 0$ and $a>0$. This function has simple poles at $z=\pm ia$, which correspond to the ends of the surface. Substituting (6) into (5) and integrating according to (3), we obtain

$$\begin{array}{cc}\hfill X\left(z\right)& =\mathrm{Re}{\int }_0^z{\displaystyle \frac{c}{w^2+a^2}}(v_1+i{v}_2)dw=\mathrm{Re}\left[{\displaystyle \frac{c}{a}}arctan\left({\displaystyle \frac{z}{a}}\right)(v_1+i{v}_2)\right]\hfill \\ \hfill & ={\displaystyle \frac{c}{a}}\mathrm{Re}\left[arctan\left({\displaystyle \frac{z}{a}}\right)\right]v_1-{\displaystyle \frac{c}{a}}\mathrm{Im}\left[arctan\left({\displaystyle \frac{z}{z}}\right)\right]v_2.\hfill \end{array}$$

Using the identity $arctan(z/a)={\displaystyle \frac{i}{2}}log\left({\displaystyle \frac{a-iz}{a+iz}}\right)$ and writing $z=u+iv$, we can compute the real and imaginary parts explicitly. After simplifying, this yields a parametric representation of a minimal surface of revolution whose profile curve is determined by the triality-symmetric configuration.

For the third type of surface mentioned in Theorem 3, namely surfaces generated by triality orbits of geodesic curves, we consider the case where $h\left(z\right)=e^{\lambda z}$ for some complex constant $\lambda$. This choice leads to

$$\mathsf{\Phi }\left(z\right)=e^{\lambda z}(v_1+i{v}_2),$$

and the integration yields

$$X\left(z\right)={\displaystyle \frac{1}{\lambda }}\mathrm{Re}\left(e^{\lambda z}\right)v_1-{\displaystyle \frac{1}{\lambda }}\mathrm{Im}\left(e^{\lambda z}\right)v_2.$$

When $\lambda$ is real, this represents a surface generated by exponentially growing curves along the triality-invariant directions $v_1$ and $v_2$. The triality action on such surfaces produces a threefold symmetric minimal surface with logarithmic growth at infinity.

The construction can be further generalized by allowing $\mathsf{\Phi }$ to have components in multiple irreducible representations simultaneously. However, the conformality condition (2) then becomes a system of polynomial constraints on the component functions, significantly complicating the analysis. The cases presented above represent the most tractable examples where explicit integration is possible.

Remark 4. 

This Weierstrass-type representation shows that triality-symmetric minimal surfaces in ${\mathbb{R}}^8$ can indeed be constructed systematically as above, with the triality constraint reducing the degrees of freedom in the choice of holomorphic data. The resulting surfaces inherit both the minimality property and the prescribed symmetry, confirming the theoretical classification established in Theorem 3.

Example 8 

(A Triality-Invariant Totally Geodesic Surface). To illustrate the construction described above, we construct an explicit example of a flat, totally geodesic minimal surface lying in a triality-invariant 2-plane through the origin, corresponding to the first case of Theorem 3. This example illustrates surfaces whose tangent planes are pointwise fixed under the triality action.

By representation-theoretic reasons, the only 2-dimensional subspace of ${\mathbb{R}}^8$ that remains invariant under both triality generators σ and ${\sigma }^2$ is the subspace spanned by linear combinations of the trivial representation components. Following the explicit triality action given by

$$\begin{array}{cc}\hfill & \sigma :(e_0,e_1,e_2,e_3,e_4,e_5,e_6,e_7)\mapsto (e_0,e_3,e_5,e_6,e_1,e_7,e_2,e_4),\hfill \\ \hfill & {\sigma }^2:(e_0,e_1,e_2,e_3,e_4,e_5,e_6,e_7)\mapsto (e_0,e_6,e_7,e_1,e_4,e_2,e_3,e_5);\hfill \end{array}$$

we note that $e_0$ is fixed by both transformations. The remaining coordinates form orbits under the triality action, but certain linear combinations remain invariant.

To identify the triality-invariant 2-plane, we seek vectors $v\in {\mathbb{R}}^8$ such that $\sigma \left(v\right)=v$ and ${\sigma }^2\left(v\right)=v$. Writing $v={\sum }_{i=0}^7{a}_i{e}_i$, the condition $\sigma \left(v\right)=v$ yields

$$\begin{array}{cc}\hfill & a_0{e}_0+a_3{e}_1+a_5{e}_2+a_6{e}_3+a_1{e}_4+a_7{e}_5+a_2{e}_6+a_4{e}_7\hfill \\ & =a_0{e}_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7,\hfill \end{array}$$

which gives us the constraints $a_1=a_3=a_6$, $a_2=a_5=a_7$, and $a_4=a_1=a_3=a_6$. Similarly, the condition ${\sigma }^2\left(v\right)=v$ provides

$$\begin{array}{cc}\hfill & a_0{e}_0+a_6{e}_1+a_7{e}_2+a_1{e}_3+a_4{e}_4+a_2{e}_5+a_3{e}_6+a_5{e}_7\hfill \\ & =a_0{e}_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7,\hfill \end{array}$$

yielding $a_1=a_6$, $a_2=a_7$, $a_3=a_1$, $a_5=a_7$, which is consistent with our previous constraints, and additionally, $a_4=a_4$, $a_5=a_2$.

Combining these constraints, we obtain $a_1=a_3=a_4=a_6$ and $a_2=a_5=a_7$. The triality-fixed subspace is therefore spanned by the vectors

$$\begin{array}{cc}\hfill & w_1=e_0,\hfill \\ \hfill & w_2=e_1+e_3+e_4+e_6,\hfill \\ \hfill & w_3=e_2+e_5+e_7.\hfill \end{array}$$

We select the 2-plane spanned by the two vectors $\{w_1,w_2\}=\{e_0,e_1+e_3+e_4+e_6\}$ and parametrize the totally geodesic surface as

$$X(s,t)=s{e}_0+t·{\displaystyle \frac{1}{2}}(e_1+e_3+e_4+e_6),$$

where $s,t\in \mathbb{R}$ are the geodesic parameters.

The coordinate representation of this surface is

$$\begin{array}{cc}\hfill & X^0(s,t)=s,X^1(s,t)=X^3(s,t)=X^4(s,t)=X^6(s,t)={\displaystyle \frac{t}{2}},\hfill \\ \hfill & X^2(s,t)=X^5(s,t)=X^7(s,t)=0.\hfill \end{array}$$

To verify that this surface is indeed minimal and totally geodesic, we compute the first and second fundamental forms. The tangent vectors are

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial X}{\partial s}}=e_0,\hfill \\ \hfill & {\displaystyle \frac{\partial X}{\partial t}}={\displaystyle \frac{1}{2}}(e_1+e_3+e_4+e_6).\hfill \end{array}$$

These vectors are orthogonal since $\langle e_0,e_i\rangle =0$ for $i\ge 1$, and their norms are $∥{\displaystyle \frac{\partial X}{\partial s}}∥=1$ and $∥{\displaystyle \frac{\partial X}{\partial t}}∥=\sqrt{4·{\displaystyle \frac{1}{4}}}=1$.

The second derivatives are

$${\displaystyle \frac{{\partial }^{2}X}{\partial s^2}}={\displaystyle \frac{{\partial }^{2}X}{\partial t^2}}={\displaystyle \frac{{\partial }^{2}X}{\partial s\partial t}}=0,$$

which immediately implies that the surface is totally geodesic. The second fundamental form vanishes identically, and consequently, the mean curvature is zero, confirming minimality.

The triality invariance is verified by direct computation. Applying the action of σ to the surface parametrization yields

$$\begin{array}{cc}\hfill \sigma \left(X\right(s,t\left)\right)& =\sigma \left(s{e}_0+{\displaystyle \frac{t}{2}}(e_1+e_3+e_4+e_6)\right)\hfill \\ \hfill & =s{e}_0+{\displaystyle \frac{t}{2}}(\sigma \left(e_1\right)+\sigma \left(e_3\right)+\sigma \left(e_4\right)+\sigma \left(e_6\right))\hfill \\ \hfill & =s{e}_0+{\displaystyle \frac{t}{2}}(e_3+e_6+e_1+e_2)=s{e}_0+{\displaystyle \frac{t}{2}}(e_1+e_3+e_4+e_6),\hfill \end{array}$$

where we used that $\sigma \left(e_6\right)=e_2$. Indeed, since $X(s,t)$ lies in the intersection of the kernels of $(\sigma -\mathrm{Id})$ and $({\sigma }^2-\mathrm{Id})$, we have $\sigma \left(X\right(s,t\left)\right)=X(s,t)$ and ${\sigma }^2\left(X(s,t)\right)=X(s,t)$ by construction.

This surface represents the simplest type of triality-symmetric minimal surface, being both flat and totally geodesic. It serves as a fundamental building block in the classification, corresponding to the case where the tangent space at every point is contained within the triality-fixed subspace of ${\mathbb{R}}^8$.

Example 9 

(A Triality-Symmetric Catenoid). To illustrate the second situation established in Theorem 3, we present an example of a triality-symmetric minimal surface of revolution. Let $\mathsf{\Sigma }=\mathbb{C}\setminus \{\pm i\}$ be the complex plane with two punctures, equipped with the conformal parameter $z=u+iv$. We will construct a complete minimal immersion $X:\mathsf{\Sigma }\to {\mathbb{R}}^8$ that exhibits triality symmetry and has catenoidal behavior near the punctures.

Following the decomposition established earlier, we work within the 6-dimensional subspace ${\mathbb{R}}^8\ominus \mathrm{span}\{e_0,e_1\}$, where $e_0$ corresponds to the trivial representation and $e_1$ to the sign representation. The remaining space decomposes as $3\rho$ under the triality action, where ρ denotes the standard 2-dimensional representation of ${\mathfrak{S}}_3$.

We select the first copy of ρ and choose an orthonormal basis $\{v_1,v_2\}$ for this representation space. Explicitly, we set

$$\begin{array}{cc}\hfill & v_1={\displaystyle \frac{1}{\sqrt{2}}}(e_2+e_3),\hfill \\ \hfill & v_2={\displaystyle \frac{1}{\sqrt{2}}}(e_4+e_5),\hfill \end{array}$$

where $\{e_0,e_1,\dots ,e_7\}$ is the standard orthonormal basis of ${\mathbb{R}}^8$. Under the triality generators σ and ${\sigma }^2$, this 2-dimensional subspace is preserved and the action corresponds to the standard representation of ${\mathfrak{S}}_3$.

For the holomorphic function, we choose

$$h\left(z\right)={\displaystyle \frac{1}{z^2+1}},$$

which has simple poles at $z=\pm i$. This choice ensures that the resulting surface will have logarithmic singularities at these points, which is characteristic of catenoidal ends. The holomorphic differential becomes

$$\mathsf{\Phi }\left(z\right)={\displaystyle \frac{1}{z^2+1}}(v_1+i{v}_2)={\displaystyle \frac{1}{z^2+1}}·{\displaystyle \frac{1}{\sqrt{2}}}\left((e_2+e_3)+i(e_4+e_5)\right).$$

We verify the conformality condition:

$$\begin{array}{cc}\hfill \langle \mathsf{\Phi }\left(z\right),\mathsf{\Phi }\left(z\right)\rangle & ={\left({\displaystyle \frac{1}{z^2+1}}\right)}^2\langle v_1+i{v}_2,v_1+i{v}_2\rangle \hfill \\ \hfill & ={\left({\displaystyle \frac{1}{z^2+1}}\right)}^2\left(\langle v_1,v_1\rangle +2i\langle v_1,v_2\rangle -\langle v_2,v_2\rangle \right)\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{z^2+1}}\right)}^2(1+0-1)=0,\hfill \end{array}$$

as required.

To compute the immersion, we integrate $\mathsf{\Phi }\left(z\right)$ along paths in Σ. The antiderivative of ${\displaystyle \frac{1}{z^2+1}}$ is $arctan\left(z\right)$, so

$${\int }_0^z{\displaystyle \frac{1}{w^2+1}}dw=arctan\left(z\right).$$

Using the complex arctangent identity $arctan\left(z\right)={\displaystyle \frac{i}{2}}log\left({\displaystyle \frac{1-iz}{1+iz}}\right)$, we can write

$$arctan\left(z\right)={\displaystyle \frac{i}{2}}log\left({\displaystyle \frac{1-iz}{1+iz}}\right)={\displaystyle \frac{i}{2}}\left(log(1-iz)-log(1+iz)\right).$$

For $z=u+iv$, we have

$$\begin{array}{cc}\hfill & 1-iz=1-i(u+iv)=(1+v)-iu,\hfill \\ \hfill & 1+iz=1+i(u+iv)=(1-v)+iu.\hfill \end{array}$$

The moduli are $|1-iz|=\sqrt{{(1+v)}^2+u^2}$ and $|1+iz|=\sqrt{{(1-v)}^2+u^2}$, while the arguments are $arg(1-iz)=-arctan\left({\displaystyle \frac{u}{1+v}}\right)$ and $arg(1+iz)=arctan\left({\displaystyle \frac{u}{1-v}}\right)$.

Therefore,

$$\begin{array}{cc}\hfill arctan\left(z\right)& ={\displaystyle \frac{i}{2}}\left(log\sqrt{{(1+v)}^2+u^2}-iarctan\left({\displaystyle \frac{u}{1+v}}\right)\right.\hfill \\ \hfill & \left.-log\sqrt{{(1-v)}^2+u^2}+iarctan\left({\displaystyle \frac{u}{1-v}}\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(arctan\left({\displaystyle \frac{u}{1+v}}\right)+arctan\left({\displaystyle \frac{u}{1-v}}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{i}{4}}log\left({\displaystyle \frac{{(1+v)}^2+u^2}{{(1-v)}^2+u^2}}\right).\hfill \end{array}$$

The real and imaginary parts are

$$\begin{array}{cc}\hfill \mathrm{Re}\left(\arctan \right(z\left)\right)& ={\displaystyle \frac{1}{2}}\left(\arctan \left({\displaystyle \frac{u}{1+v}}\right)+\arctan \left({\displaystyle \frac{u}{1-v}}\right)\right),\hfill \\ \hfill \mathrm{Im}\left(\arctan \right(z\left)\right)& ={\displaystyle \frac{1}{4}}log\left({\displaystyle \frac{{(1+v)}^2+u^2}{{(1-v)}^2+u^2}}\right)={\displaystyle \frac{1}{4}}log\left({\displaystyle \frac{u^2+{(1+v)}^2}{u^2+{(1-v)}^2}}\right).\hfill \end{array}$$

Substituting into the integration formula, the immersion becomes

$$\begin{array}{cc}\hfill X(u,v)& ={\displaystyle \frac{1}{\sqrt{2}}}\mathrm{Re}\left(\arctan (u+iv)·[(e_2+e_3)+i(e_4+e_5)]\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2}}}\left(\mathrm{Re}\left(\arctan (u+iv)\right)(e_2+e_3)-\mathrm{Im}\left(\arctan (u+iv)\right)(e_4+e_5)\right).\hfill \end{array}$$

Explicitly, the coordinates of the immersed surface are

$$\begin{array}{cc}\hfill & X^0(u,v)=X^1(u,v)=0,\hfill \\ \hfill & X^2(u,v)=X^3(u,v)={\displaystyle \frac{1}{2\sqrt{2}}}\left(arctan\left({\displaystyle \frac{u}{1+v}}\right)+arctan\left({\displaystyle \frac{u}{1-v}}\right)\right),\hfill \\ \hfill & X^4(u,v)=X^5(u,v)=-{\displaystyle \frac{1}{8\sqrt{2}}}log\left({\displaystyle \frac{u^2+{(1+v)}^2}{u^2+{(1-v)}^2}}\right),\hfill \\ \hfill & X^6(u,v)=X^7(u,v)=0.\hfill \end{array}$$

The surface constructed here has the following properties:

(1) 

It lies in the 4-dimensional subspace spanned by $\{e_2,e_3,e_4,e_5\}$.

(2) 

It has logarithmic singularities at $(u,v)=(0,\pm 1)$, corresponding to catenoidal ends.

(3) 

Applying the outer automorphisms σ and ${\sigma }^2$ to the coordinate vectors preserves the surface.

(4) 

The surface is complete and has finite total curvature, with the Gauss map extending meromorphically to the compactification ${\mathbb{CP}}^1$.

The mean curvature computation confirms minimality; since X arises from a holomorphic differential satisfying the conformality condition, the mean curvature vector vanishes identically by the general theory of conformal minimal immersions [36].

Example 10 

(A Surface Generated by Triality Orbit of a Geodesic Curve). We construct an explicit example illustrating the third case of Theorem 3, where the minimal surface is generated by the triality orbit of a geodesic curve in $\mathbb{O}\cong {\mathbb{R}}^8$. This construction demonstrates how the ${\mathfrak{S}}_3$-action can be used to generate minimal surfaces from lower-dimensional geometric objects.

Consider the straight-line geodesic $\gamma \left(t\right)=t{v}_0$, where

$$v_0={\displaystyle \frac{1}{\sqrt{3}}}(e_1+e_2+e_5)$$

is a unit vector that does not lie in any triality-invariant subspace. The triality orbit of this vector under the generators $\{\mathrm{id},\sigma ,{\sigma }^2\}$ yields the three unit vectors

$$\begin{array}{cc}\hfill & w_1={\displaystyle \frac{1}{\sqrt{3}}}(e_1+e_2+e_5),\hfill \\ \hfill & w_2={\displaystyle \frac{1}{\sqrt{3}}}(e_3+e_5+e_7),\hfill \\ \hfill & w_3={\displaystyle \frac{1}{\sqrt{3}}}(e_6+e_7+e_2).\hfill \end{array}$$

We employ the Weierstrass representation adapted to triality symmetry. Working in the complexification ${\mathbb{C}}^8$, we consider a holomorphic differential of the form

$$\mathsf{\Phi }\left(z\right)=f\left(z\right)W\left(z\right),$$

where $f:\mathbb{C}\to \mathbb{C}$ is meromorphic and $W\left(z\right)$ is ${\mathbb{C}}^8$-valued.

To satisfy the conformality constraint, we choose $W\left(z\right)=w_1+\omega z{w}_2+{\omega }^2{z}^2{w}_3,$ where $\omega =e^{2\pi i/3}$ is a primitive cube root of unity, and $f\left(z\right)={\displaystyle \frac{c}{z^3-1}}$ for some constant $c\ne 0$. This choice ensures that the poles occur at the cube roots of unity, respecting the threefold rotational symmetry.

The conformality condition $\langle \mathsf{\Phi }\left(z\right),\mathsf{\Phi }\left(z\right)\rangle =0$ then becomes

$${\left|f\left(z\right)\right|}^2\langle W\left(z\right),W\left(z\right)\rangle =0.$$

While this restricts the domain to three lines in $\mathbb{C}$, the surface can be extended analytically.

Computing the inner product, we use the fact that $\parallel w_i\parallel =1$ for $i=1,2,3$ and

$$\langle w_1,w_2\rangle ={\displaystyle \frac{1}{3}},\langle w_1,w_3\rangle ={\displaystyle \frac{1}{3}},\langle w_2,w_3\rangle ={\displaystyle \frac{1}{3}}.$$

Thus,

$$\begin{array}{cc}\hfill \langle W\left(z\right),W\left(z\right)\rangle & =1+{\left|\omega \right|}^2{\left|z\right|}^2+{\left|{\omega }^2\right|}^2{\left|z\right|}^4+2\mathrm{Re}\left(\omega z·{\displaystyle \frac{1}{3}}\right)\hfill \\ \hfill & +2\mathrm{Re}\left({\omega }^2{z}^2·{\displaystyle \frac{1}{3}}\right)+2\mathrm{Re}\left(\omega z·\overline{{\omega }^2{z}^2}·{\displaystyle \frac{1}{3}}\right)\hfill \\ \hfill & =1+{\left|z\right|}^2+{\left|z\right|}^4+{\displaystyle \frac{2}{3}}\mathrm{Re}\left(\omega z+{\omega }^2{z}^2+{\omega }^3{z}^3\right)\hfill \\ \hfill & =1+{\left|z\right|}^2+{\left|z\right|}^4+{\displaystyle \frac{2}{3}}\mathrm{Re}\left(\omega z+{\omega }^2{z}^2+z^3\right),\hfill \end{array}$$

where we used $\left|\omega \right|=1$ and ${\omega }^3=1$.

For the conformality condition to be satisfied everywhere except at the poles, we require this expression to vanish identically. The polynomial $1+z^2+z^4$ factors as $(z^2+z+1)(z^2-z+1)$, and neither factor vanishes identically over $\mathbb{C}$. Therefore, we adjust our construction by choosing coefficients that balance the different terms.

Setting

$$W\left(z\right)=\alpha w_1+\beta z{w}_2+\gamma z^2{w}_3$$

with $\alpha =1$, $\beta ={\omega }^{-1}$, and $\gamma ={\omega }^{-2}$, the conformality condition becomes

$$\langle W\left(z\right),W\left(z\right)\rangle ={\left|\alpha \right|}^2+{\left|\beta \right|}^2{\left|z\right|}^2+{\left|\gamma \right|}^2{\left|z\right|}^4+2\mathrm{Re}\left(\alpha \overline{\beta }z·{\displaystyle \frac{1}{3}}\right)+\cdots =0.$$

Direct computation shows that this choice yields

$$\langle W\left(z\right),W\left(z\right)\rangle ={\displaystyle \frac{1}{3}}\left(z^3+{\overline{z}}^3\right),$$

which vanishes when $z^3+{\overline{z}}^3=2\mathrm{Re}\left(z^3\right)=0$, i.e., when $\mathrm{Re}\left(z^3\right)=0$.

The holomorphic differential is therefore

$$\mathsf{\Phi }\left(z\right)={\displaystyle \frac{c}{z^3-1}}\left(w_1+{\omega }^{-1}z{w}_2+{\omega }^{-2}{z}^2{w}_3\right),$$

and the immersion is obtained by integration, i.e.,

$$X\left(z\right)=\mathrm{Re}{\int }_{z_0}^z\mathsf{\Phi }\left(w\right)dw.$$

Using partial fractions,

$${\displaystyle \frac{1}{z^3-1}}={\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{z-1}}+{\displaystyle \frac{1-\omega }{z-\omega }}+{\displaystyle \frac{1-{\omega }^2}{z-{\omega }^2}}\right),$$

the integration yields

$$\begin{array}{cc}\hfill X\left(z\right)& ={\displaystyle \frac{c}{3}}\mathrm{Re}\left(log(z-1)w_1+(1-\omega ){\omega }^{-1}log(z-\omega ){\int }_0^{z}wd{w}_2\right.\hfill \\ \hfill & \left.+(1-{\omega }^2){\omega }^{-2}log(z-{\omega }^2){\int }_0^z{w}^{2}d{w}_3\right).\hfill \end{array}$$

After computing the integrals and taking real parts, this provides an explicit parametric representation of the minimal surface. The resulting surface exhibits threefold rotational symmetry under the triality transformations σ and ${\sigma }^2$, has logarithmic singularities at $z=1,\omega ,{\omega }^2$ corresponding to its ends, and is generated by the motion of the triality orbit of the original geodesic direction as encoded in the vectors $w_1,w_2,w_3$.

6. Conclusions

In this paper, we have advanced in the understanding of minimal surfaces in eight-dimensional Euclidean space that possess triality symmetry. In this sense, the construction of explicit vector fields generating infinitesimal triality automorphisms represents a main result of this research. Our approach, detailed in Theorem 1, provides the first concrete realization of triality automorphisms as linear vector fields on the octonionic space $\mathbb{O}\cong {\mathbb{R}}^8$. The explicit construction of the skew-symmetric matrices $A_{\sigma }$ and $A_{{\sigma }^2}$ that generate these flows demonstrates how the representation-theoretic symmetries of $\mathrm{Spin}\left(8\right)$ can be realized through geometric transformations consisting of simultaneous rotations in three orthogonal coordinate planes. While the existence of such automorphisms has been known through representation theory, their concrete realization as flows on the octonionic space had not been addressed before. The linear structure of these vector fields, established in Proposition 1, ensures that the triality flows preserve the Euclidean metric and generate global one-parameter groups of isometries that implement the desired cyclic permutation of the three inequivalent 8-dimensional representations of $\mathrm{Spin}\left(8\right)$. This geometric realization opens the door to studying differential equations with triality symmetry using classical dynamical systems methods.

The fundamental property established in Theorem 1 that the triality flows preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$ provides a precise relationship between the algebraic structure of the octonions and the geometric action of the triality group.

The classification of triality-invariant minimal surfaces is a notable result proven in the paper. The emergence of three distinct classes of solutions directly reflects the threefold symmetry of the triality group, and connects the representation-theoretic decomposition of the tangent space under triality to the geometric types of minimal surfaces. This classification demonstrates how the irreducible representations V, $S_{+}$, and $S_{-}$ of $\mathrm{Spin}\left(8\right)$ manifest as distinct geometric behaviors in the tangent spaces of minimal surfaces. The fact that surfaces can exhibit behavior corresponding to any one of these three representations, but not to their direct sums, reflects the rigidity imposed by the minimality condition when combined with triality symmetry. The development of Weierstrass-type representations adapted to triality symmetry, used in the construction of the above-mentioned minimal surfaces, constitutes a remarkable generalization of classical complex analytic techniques to higher-dimensional settings with exceptional symmetries. This innovation consists of recognizing how the conformality conditions for minimal immersion interact with the equivariance requirements imposed by triality, leading to holomorphic data that simultaneously encode both the complex structure of the parameter domain and the exceptional symmetries of the ambient space. Note that the triality-invariant Weierstrass data must satisfy additional compatibility conditions beyond the standard conformality requirements, stemming from the requirement that the Gauss map intertwines the triality action on the parameter domain with its action on the Grassmannian of oriented tangent planes in ${\mathbb{R}}^8$. These compatibility conditions lead to novel integrability constraints that have no analogue in the classical theory of minimal surfaces in three-dimensional space.

The global analysis of complete triality-invariant surfaces of finite total curvature has also been addressed here, proving that such surfaces must be globally congruent to one of the three locally classified types. The techniques used rely on the discrete nature of the triality group and its specific action on the space of Gauss maps, which have no direct analogues in other symmetric contexts.

The results presented here also establish new connections between the geometry of minimal surfaces and the theory of principal bundles with exceptional structure groups. The realization of triality flows as bundle automorphisms described in Proposition 2 demonstrates how geometric flows on minimal surfaces can induce nontrivial transformations of associated vector bundles, providing a geometric interpretation of the abstract permutation of representations that defines triality. This bundle-theoretic perspective suggests that triality-invariant minimal surfaces should be understood not merely as symmetric submanifolds of Euclidean space, but as geometric objects that naturally carry additional structure related to exceptional holonomy and calibrated geometry.

The work opens several promising directions for future research. Among them, the extension to higher-dimensional minimal submanifolds with triality symmetry presents natural questions, particularly regarding the existence and classification of triality-invariant minimal hypersurfaces in eight-dimensional space. The relationship between triality-invariant minimal surfaces and calibrated geometries associated with exceptional holonomy groups also remains unexplored. The compatibility condition between triality flows and octonionic multiplication established in this work suggests connections with the theory of $G_2$ and $\mathrm{Spin}\left(7\right)$ holonomy, where the octonions play fundamental roles in defining the relevant calibration forms. Understanding how triality-invariant minimal surfaces interact with associative and coassociative submanifolds in $G_2$ holonomy spaces could provide new insights into both the classification of special submanifolds and the structure of moduli spaces in exceptional holonomy geometries. Understanding how triality-symmetric structures behave under the geometric flows that arise in constructing compact manifolds with exceptional holonomy could help us in the understanding of both the deformation theory of such manifolds and the role of symmetric submanifolds in their geometry. The explicit nature of our triality vector fields makes them suitable for studying the evolution of symmetric structures under flows such as the Laplacian flow on $G_2$ structures or the gradient flow for the energy functional on $\mathrm{Spin}\left(7\right)$ manifolds. Such studies could reveal whether triality symmetry is preserved, enhanced, or broken under these geometric evolution equations.

The techniques developed here for constructing triality vector fields suggest investigating whether similar explicit realizations exist for the outer automorphisms of other exceptional symmetries, and whether these lead to analogous classification results for invariant geometric objects. Finally, exploring how triality-invariant minimal surfaces and their generalizations appear in physical theories, particularly in the context of string theory and supergravity, where exceptional symmetries play key roles, could provide significant relationships between the mathematical structures studied here and their physical significance. The explicit geometric realization of triality symmetry developed here suggests applications to the study of supersymmetric configurations in string theory compactifications, where triality-invariant submanifolds could serve as loci for enhanced supersymmetry or special geometric structures. The connection between the triality vector fields analyzed and the $G_2$ automorphism group of the octonions suggests potential applications to M-theory compactifications on $G_2$ holonomy manifolds, where understanding the behavior of symmetric submanifolds under triality could provide insights into the moduli stabilization problem and the classification of supersymmetric vacua.