Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine, to analyse geometrically, and to classify up to Lie algebra automorphisms, in a relatively easy manner, coboundary Liebialgebras on real four-dimensional indecomposable Lie algebras. The Darboux family notion can be consideredas a generalisation of the Darboux polynomial for a vector field. The classification of $r$-matrices and solutions to classical Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.


Introduction
Lie bialgebras [15,17,18,29] appeared as a tool to study integrable systems [20,21]. A Lie bialgebra is a Lie algebra g along with a Lie bracket on its dual space g * that amounts to a cocycle in a Chevalley-Eilenberg cohomology of g. Lie bialgebras have also applications to quantum gravity [4,8,38,39] and other research fields [15,29,43]. In differential geometry, they occur in the problem of classifying Poisson Lie groups [3,15].
Although much research has been devoted to the classification of Lie bialgebras up to Lie algebra automorphisms, there are still many open problems. More specifically, Lie bialgebras on two-and three-dimensional Lie algebras g have been completely classified [22,25]. Particular instances of Lie bialgebras with dim g > 3, e.g. for a semi-simple g, have also been studied [2,11,14,30,36,43,44]. Employed techniques are rarely non-algebraic (cf. [35]) and they are not very effective to analyse Lie bialgebras when dim g > 3 [2,15,22,25,35]. This motivates the search for new approaches to the study and classification of Lie bialgebras [35]. In particular, we are here interested in geometric approaches.
A coboundary Lie bialgebra is a particular type of Lie bialgebra that is characterised by means of an r-matrix, namely a bivector on a Lie algebra g that is a solution to its modified classical Yang-Baxter equation (mCYBE) [15,25]. This work introduces novel geometric techniques to classify, up to Lie algebra automorphisms, coboundary Lie bialgebras and their r-matrices on a fixed Lie algebra g.
In particular, we introduce a generalisation, the so-called Darboux family, of the notion of Darboux polynomial for a polynomial vector field. On a finite-dimensional vector space E, a polynomial function is a polynomial expression on a set of linear coordinates on E. A Darboux polynomial, P , for a polynomial vector field X on E is a polynomial function on E so that XP = f P for a certain polynomial f on E, the so-called cofactor of P relative to X [16,33]. It is worth noting that, geometrically, one cannot define intrinsically what a polynomial on a general manifold is, as the explicit form of a function on a manifold depends on the chosen coordinate system.
More generally, let us consider a q-dimensional Lie algebra of vector fields, let us say V := X 1 , . . . , X q , on a manifold M . A Darboux family for V is an s-dimensional vector space A := f 1 , . . . , f s of smooth functions on M such that X α f β = s γ=1 g αβγ f γ for certain smooth functions g αβγ ∈ C ∞ (M ) with α = 1, . . . , q and β, γ = 1, . . . , s. In this work, Darboux families are studied and employed to classify up to Lie algebra automorphisms the r-matrices and coboundary Lie bialgebras on real four-dimensional Lie algebras that are indecomposable [50], i.e. they cannot be written as a direct sum of two proper ideals, namely ideals that are different of the zero and the total Lie algebra. Nevertheless, our methods can also be applied to any other Lie algebra.
Let Aut(g) stand for the Lie group of Lie algebra automorphisms of a Lie algebra g. Then, Aut(g) naturally acts on the space Λ 2 g of bivectors [1] of g and, more specifically, on the space Y g ⊂ Λ 2 g of solutions to the mCYBE on g (see [15,35]). The classes of equivalent r-matrices (up to Lie algebra automorphisms of g) are given by the orbits of the action of Aut(g) on Y g . Characterising and analysing such orbits is in general complicated even when the form of Aut(g) is explicitly known (see [22] for a typical algebraic approach to Lie bialgebras on three-dimensional Lie algebras). Most techniques in the literature are algebraic [22,25]. Instead, we use here a more geometrical approach. Let us sketch our main ideas. We use a simple method (see Proposition 2.2 and Remark 2.1) to determine the Lie algebra of fundamental vector fields, V g , of the action of Aut(g) on Λ 2 g. Our technique does not require to know the explicit form of Aut(g). Instead, it is enough to know the space of derivations on g, which can be derived by solving a linear system of algebraic equations. We show that the orbits of the connected part of Aut(g) containing its neutral element, Aut c (g), are the integral connected submanifolds of the generalised distribution, E g , spanned by the vector fields of V g (see [27,31,41,48,49,51] for general results on generalised distributions). In fact, as V g is a finite-dimensional Lie algebra, E g is integrable [31]. Finding the integral connected submanifolds of a generalised distribution, its so-called strata, is more complicated than finding the leaves of standard distributions because strata cannot always be determined by a family of common first-integrals for the elements of V g , as it happens in the case of distributions (cf. [1,28]). We determine them here via our Darboux families, which is relatively easy as illustrated by our examples. Once the strata of E g in Y g have been obtained, the use of the action on such strata of an element of each connected part of Aut(g) allows us to determine the orbits of Aut(g) on Y g , which gives us the desired classification of r-matrices. Note that our method do rely on the determination of a single element of each connected component of Aut(g) and, more relevantly, it does not need the explicit action of the whole Aut(g) on Λ 2 g.
It is known that two r-matrices for g that are not equivalent up to a Lie algebra automorphism may give rise to coboundary Lie bialgebras that are equivalent up to a Lie algebra automorphism (cf. [15,35]). Let (Λ 2 g) g be the space of bivectors of g that are invariant relative to the action of elements of g via the algebraic Schouten bracket [15,35]. In our work, we prove that the orbits of Aut(g) on Y g that project onto the same orbit of the natural action of Aut(g) on Λ 2 g/(Λ 2 g) g (see [35]) are exactly the r-matrices that lead to equivalent (up to Lie algebra automorphisms) coboundary Lie bialgebras on g.
Our methods are computationally affordable for the study and classification of general coboundary Lie bialgebras with three-and four-dimensional g. Our techniques are quite probably appropriate for looking into and classifying Lie bialgebras on any five-dimensional g and, possibly, for other particular instances of higher-dimensional Lie algebras g. Indeed, our procedures lead to the classification of real coboundary Lie bialgebras, up to Lie algebra automorphisms, on any four-dimensional indecomposable Lie algebra g (see [50] for a classification of indecomposable Lie algebras up to dimension six). Our results are summarised in Table 2. This is a remarkable advance relative to other classification works in the literature [2,22,25]. Indeed, the most complicated classification in the literature so far is probably the classification of Lie bialgebras on symplectic four-dimensional Lie algebras [2]. Additionally, other four-dimensional Lie bialgebras have been partially studied in the mathematics and physics literature (see [4,6,7,10,12,11,30,43,44] and references therein). As a byproduct of our research, a method for the matrix representation of a class of finite-dimensional Lie algebras with a non-trivial center, which cannot be represented through the matrices of the adjoint representation, is given. This is interesting not only for our purposes, but also in many other works where such a representation is employed in practical calculations, e.g. [5].
The structure of the paper goes as follows. Section 2 surveys the main notions on Lie bialgebras and their derivations, g-modules, and the notation to be used. In Section 3, the theory of generalised distributions is discussed. A method to obtain a matrix representation of a class of Lie algebras with non-trivial center is given in Section 4. Section 5 introduces Darboux families and shows how they can be used to classify r-matrices on the Lie algebra s 4,1 (according toŠnobl and Winternitz's notation in [50]), up to Lie algebra automorphisms thereof. Section 6 analyses several geometric properties of solutions to mCYBEs and it also provides some hints on the use Darboux families to study the equivalence up to Lie algebra automorphisms of r-matrices and coboundary Lie bialgebras. In Section 7, Darboux families are applied to studying and classifying coboundary Lie bialgebras on real four-dimensional indecomposable Lie algebras. Section 8 summarises our results and presents some further work in progress.

Fundamentals on Lie bialgebras and their derivations
Let us provide a brief account on the notions of Lie bialgebras, Schouten brackets, and g-modules to be used hereafter (see [15,29,51] for further details). Some simple results on the geometric properties of the action of Aut(g) on spaces of k-vectors are provided. Our approach is more geometric than in standard works on Lie bialgebras. We hereafter assume that g and E are a finite-dimensional real Lie algebra and a finite-dimensional real vector space, respectively. Meanwhile, GL(E) and gl(E) stand for the Lie group of automorphisms and the Lie algebra of endomorphisms on E, respectively.
Let V m M be the vector space of m-vector fields on a manifold M . The Schouten-Nijenhuis bracket [37,51] on VM := ⊕ m∈Z V m M is the unique bilinear map [·, ·] : VM × VM → VM satisfying that where f, g ∈ C ∞ (M ), s, l ∈ N, the X 1 , . . . , X s , Y 1 , . . . , Y l are arbitrary vector fields on M , an omitted vector field X is denoted by the hat symbol X, and [X i , Y j ] is the Lie bracket 1 of X i and Y j (see [37] for more details). Remarkably, [X , Y] ∈ V s+l−1 for X ∈ V s M and Y ∈ V l M . The space, V L G, of left-invariant elements of VG for a Lie group G is closed relative to [·, ·]. In particular, left-invariant vector fields on G, i.e. V L 1 G, span a finite-dimensional Lie algebra called the Lie algebra of G, which can be identified with T e G [1]. Vice versa, every abstract finitedimensional Lie algebra g can be thought of as the Lie algebra of left-invariant vector fields of a Lie group [19]. Meanwhile, V L G can be identified with the Grassmann algebra Λg, namely the algebra relative to the exterior product spanned by all the multivectors of the Lie algebra, g, of G [26]. Moreover, [·, ·] can be restricted to V L G leading to the algebraic Schouten bracket on Λg [35,51]. For simplicity, we will call it the Schouten bracket on Λg.
Recall that the Grassmann algebra ΛE of a vector space E satisfies that ΛE = m∈Z Λ m E, where Λ m E is the vector space of m-vectors of E.
A Lie bialgebra is a pair (g, δ), where g admits a Lie bracket [·, ·] g , whilst δ : g → Λ 2 g, the cocommutator, is a linear map, its transpose δ * : Λ 2 g * → g * is a Lie bracket on g * , and A Lie bialgebra homomorphism is a Lie algebra homomorphism φ : g → h between Lie bialgebras (g, δ g ) and for every v ∈ g and some r ∈ Λ 2 g, a so-called r-matrix. To characterise r-matrices, we use the following notions and Theorem 2.1. The standard identification of an abstract Lie algebra g with the Lie algebra of left-invariant vector fields on a Lie group G allows us to understand the tensor algebra T(g) of g as the tensor algebra, T L (G), of left-invariant tensor fields on G. This gives rise to a Lie algebra representation ad : v ∈ g → ad v ∈ gl (T(g)), where ad v (w) := L v w for every w ∈ T(g) and L v w is the Lie derivative of w relative to v, which are understood as elements in T L (G) in the natural way. Note that the geometric notation L v w is conciser than algebraic ones (cf. [15]). An element q ∈ T(g) is called g-invariant if L v q = 0 for all v ∈ g. We write (T(g)) g for the set of g-invariant elements of T(g). The map ad : g → gl(T(g)) admits a restriction ad : g → gl(Λ m g). We recall that (Λ m g) g stands for the space of g-invariant m-vectors. Let us recall the following well-known result.
We call [r, r] ∈ (Λ 3 g) g the modified classical Yang-Baxter equation (mCYBE) of g, while [r, r] = 0 is referred to as the classical Yang-Baxter equation (CYBE) of g and its solutions amount to left-invariant Poisson bivectors on any Lie group G with Lie algebra isomorphic to g [51].
Note that two r-matrices r 1 , r 2 ∈ Λ 2 g satisfy that δ r1 = δ r2 if and only if r 1 − r 2 ∈ (Λ 2 g) g . Then, what really matters to the determination of coboundary Lie bialgebras is not r-matrices, but their equivalence classes in the quotient space Λ 2 is a Lie algebra morphism. A g-module (V, ρ) will be represented just by V , while ρ v (x), for any v ∈ g and x ∈ V , will be written simply as vx if ρ is understood from the context. If ad : v ∈ g → [v, ·] g ∈ gl(g) stands for the adjoint representation of g, then the fact that each [v, ·] g , with v ∈ g, is a derivation of the Lie algebra g (relative to its Lie bracket [·, ·] g ) allows us to ensure that (g, ad) is a g-module [24]. The map ad can be considered as a mapping of the form ad : g → der(g), where der(g) is the Lie algebra of derivations on g. As a second relevant example of g-module, consider the Lie group Aut(g) of the Lie algebra automorphisms of g (see [45] for details on its Lie group structure) and its Lie algebra, which is denoted by aut(g). The tangent map at the identity map on g, let us say id g ∈ Aut(g), to the injection ι : Aut(g) ֒→ GL(g) induces a Lie algebra morphism ad : aut(g) ≃ T id g Aut(g) → gl(g) ≃ T id g GL(g) and (g, ad) becomes an aut(g)-module.
In view of the properties of the algebraic Schouten bracket, each g gives rise to a g-module (Λg, ad), where ad : v ∈ g → [v, ·] ∈ gl(Λg) (cf. [51]). This fact can be viewed as a consequence of [35,Proposition 2.1]. To grasp this result and related ones, recall that every T ∈ gl(E) gives rise to the mappings Λ m T : λ ∈ Λ m E → 0 ∈ Λ m E for m ≤ 0, and the maps Λ m T ∈ gl(Λ m E), for m > 0, given by the restriction to Λ m E of Λ m T := where id is the identity on E. Moreover, ΛT := m∈Z Λ m T ∈ gl(ΛE). If T is considered as an element of GL(E), we define Λ m T := T ⊗. . .⊗T (m−operators) for m ≥ 1 and Λ m T is the identity on Λ m E for m ≤ 0. Finally, ΛT := m∈Z Λ m T .
The Lie group Aut(g) gives rise to a Lie group action T ∈ Aut(g) → Λ m T ∈ GL(Λ m g). Moreover, this gives rise to an infinitesimal Lie group action ρ : aut(g) → gl(Λ m g) such that We write V g for the Lie algebra of fundamental vector fields of this Lie group action for m = 2. Recall that aut(g) is indeed the space of derivations on g (cf. [35]). The following result was proved in [35].
In this work, we will use the following rather straightforward generalisation of Proposition 2.1.
Proof. The orbit of w ∈ Λ m g relative to Aut(g) is given by the points (Λ m T )(w) for every T ∈ Aut(g). Define exp(td) =: T t , with t ∈ R, for d ∈ der(g). Then, the tangent space at w of O w is spanned by the tangent vectors where G w is the isotropy group of w ∈ Λ m g relative to the action of Aut(g) and g w is the Lie algebra of G w . Moreover, g w is given by those d ∈ der(g) such that (Λ m d)(w) = 0. This amounts Remark 2.1. As a byproduct, Proposition 2.2 shows that the fundamental vector fields of the natural Lie group action of Aut(g) on Λ 2 g are spanned by . , x s is any linear coordinate system on Λ 2 g, the w is any point in Λ 2 g, and v belongs to der(g). Moreover, the coordinates of such vector fields are the coordinates of the extensions to Λ 2 g of the derivations of g, which can easily be obtained as

Generalised distributions
In this work we want to show that the problem of determining equivalent r-matrices up to Lie algebra automorphisms can be significantly simplified by studying the strata of the so-called generalised distributions [31,48,49,51]. Let us detail some useful facts on these geometric entities. Unless otherwise stated, we assume all objects to be smooth and globally defined. Hereafter, X(M ) stands for the Lie algebra of vector fields on M .
A generalised distribution (also called a Stefan-Sussmann distribution) on a manifold M is a correspondence D attaching each x ∈ M to a subspace D x ⊂ T x M . We call rank of D at x the dimension of D x . A generalised distribution need not have the same rank at every point of M . If D has the same rank at every point of M , then D is said to be regular or D is simply called a distribution. Otherwise, D is said to be singular. A generalised distribution D on M is involutive if every two vector fields taking values in D satisfy that their Lie bracket takes values in D as well.
A stratification, let us say F , on a manifold M is a partition of M into connected disjoint immersed submanifolds {F k } k∈I , where I is a certain set of indices, i.e. M = k∈I F k and the submanifolds {F k } k∈I satisfy F k ∩ F k ′ = ∅ for k = k ′ and k, k ′ ∈ I. The connected immersed submanifolds F k , with k ∈ I, are called the strata of the stratification. A stratification is regular if its strata are immersed submanifolds of the same dimension, whilst it is singular otherwise. Regular stratifications are called foliations and their strata are called leaves. The tangent space to a stratum, F k , of a stratification passing through a point x ∈ M is a subspace D x ⊂ T x M . All the subspaces D x ⊂ T x M for every point x ∈ M give rise to a generalised distribution D := x∈M D x on M . All the leaves of a foliation have the same dimension and, therefore, the generalised distribution formed by the tangent spaces at every point to its leaves is regular. Meanwhile, a singular stratification gives rise to a singular generalised distribution.
In this work, we are specially interested in generalised distributions generated by finite-dimensional Lie algebras of vector fields, the so-called Vessiot-Guldberg Lie algebras [34]. More specifically, let V be a Vessiot-Guldberg Lie algebra, the vector fields of V span a generalised distribution D V given by Since the space of vector fields tangent to the strata of a stratification are closed under Lie brackets, the Lie bracket of vector fields on M taking values in a distribution D can be restricted to each one of its strata. A generalised distribution D on M is integrable if there exists a stratification F on M such that each stratum F k thereof satisfies T x F k = D x for every x ∈ F k . A relevant question is whether a generalised distribution on M is integrable of not. For regular distributions, the Frobenius theorem holds [23,31].
Involutivity is a natural necessary condition for a generalised distribution to be integrable because the set of vector fields tangent to any stratum of a stratification is involutive. Nevertheless, if a generalised distribution is not regular, its involutiveness does not necessarily implies its integrability. If a generalised distribution D on M is analytical, i.e. for every x ∈ M there exists a family of analytical vector fields taking values in D and spanning D x ′ for every x ′ in an open neighbourhood of x, one has the following proposition.
Theorem 3.2. (Nagano [41] and [31]) Let M be a real analytic manifold, and let V be a sub-Lie algebra of analytic vector fields on M . Then, the induced analytic distribution D V is integrable.
If a generalised distribution is not analytical, the Stefan-Sussmann's theorem establishes additional conditions to involutivity to ensure integrability. We are more interested in the following result that ensures integrability in a particular case of relevance to us. Theorem 3.3. (Hermann [27] and [31]) Let M be a smooth manifold. If V is a finite-dimensional Lie subalgebra of X(M ), then the distribution D V is integrable.
One relevant problem concerns the determination of the form of the strata of a stratification. If a stratification is regular on an open subset U ⊂ M , then one can define locally a set of functionally independent functions whose level sets are the strata of the stratification. Indeed, this amounts to the integration of a standard distribution. If a stratification is singular, the previous statement is not longer true and other methods are needed. The following facts are useful to understand further parts of this work. First, given a point x ∈ M and a Vessiot-Guldberg Lie algebra V on M , the stratum F k of the generalised distribution D V passing through x is given by the points of the form [48] exp( where s is any natural number, X 1 , . . . , X s are any vector fields of V , and each exp(X) stands for the local diffeomorphism on M induced by the vector field X on M . Let f be a function on M such that if X ∈ V , then there exists a function f X ∈ C ∞ (M ) such that Xf = f X f . This can be considered as a not necessarily polynomial analogue of a Darboux function for a vector field. Let us assume that f = 0 at a point of F k . It immediately follows from (3.1) that f vanishes on the whole F k . Hence, two points x 1 , x 2 ∈ M such that f (x 1 ) = f (x 2 ) cannot belong to the same F k . As shown in following sections, f does not need to be a constant of motion of the vector fields in V . This is specially relevant when determining the strata of an integrable singular generalised foliation, as their strata cannot always be determined locally as the zeroes of a family of common first-integrals of the vector fields of V . Meanwhile, this can always be achieved locally for the leaves of an integrable distribution [1,51].

Obtaining a matrix representation for a Lie algebra with nontrival center
If a Lie algebra g has a nontrivial center, then the image of the map ad : g ∋ v → [v, ·] ∈ gl(g) is a matrix Lie algebra that is not isomorphic to g. Hence, the image of ad does not give a matrix algebra representation of g, which may give rise to a problem as matrix Lie algebra representation are quite practical in computations [5]. The aim of this section is to provide a method to obtain a matrix Lie algebra isomorphic to g when its center, Z(g), is not trivial, namely Z(g) = 0, and g satisfy some additional conditions. Our method is to be employed in the rest of our work during calculations.
Let {e 1 , . . . , e s , e s+1 , . . . , e n } be a basis of g such that {e 1 , . . . , e s } form a basis for Z(g). Let c k ij , with i, j, k = 1, . . . , n, be the structure constants of g in the given basis, i.e. [e i , e j ] = n k=1 c k ij e k for i, j = 1, . . . , n. Define theng := e 1 , . . . , e n , e . Let us determine the conditions ensuring thatg is a Lie algebra relative to a Lie bracket, [·, ·]ḡ, that is an extension of the Lie bracket on g, i.e. [e ′ , e ′′ ]ḡ = [e ′ , e ′′ ] g for every e ′ , e ′′ ∈ g, and [e, e i ]ḡ = α i e i for i = 1, . . . , n so that α 1 · α 2 · . . . · α s = 0. (4.1) The meaning of the last condition will be clear in a while. If [·, ·]ḡ is to be a Lie bracket, its Jacobi identity leads to the following conditions  Thus, previous conditions can be reduced to requiring that, for those indices i, j, k satisfying c k ij = 0, one gets If the latter condition is satisfied, a new Lie algebrag arises. Note that g is an ideal ofg. This leads to a Lie algebra morphism Then, condition (4.1) yields λ 1 = . . . = λ s = 0 and it turns out that v = 0. Hence, the elements R v , with v ∈ g, span a matrix Lie algebra of endomorphisms ong isomorphic to g.
It is clear that conditions (4.1) and (4.2) do not need to be satisfied for a general Lie algebra g with a nontrivial center. Nevertheless, we used in this work that this method works for all indecomposable real four-dimensional Lie algebras with non-trivial center (see [50] for a complete list of such Lie algebras). This will be enough for our purposes.
5 Darboux families and the Lie algebra s 1 Let us classify solutions to the mCYBE for the Lie algebra s 1 up to its Lie algebra automorphisms. As a byproduct, we introduce a generalisation of the concept of Darboux polynomial of a vector field.
The space of derivations on s 1 can straightforwardly be obtained. It is indeed the solution of a linear algebra problem that can be easily solved by hand calculation and/or via any mathematical computation program. The same could be achieved for any four, five, or even some other higher-dimensional Lie algebra. In particular, derivations on s 1 take the form in the basis {e 1 , e 2 , e 3 , e 4 } appearing in Table 1. By Proposition 2.2 and, more specifically, Remark 2.1, the Lie algebra V s1 of fundamental vector fields of the natural Lie group action of Aut(s 1 ) on Λ 2 s 1 is spanned by the basis (over the reals) As X 1 , . . . , X 6 close on a finite-dimensional Lie algebra, Theorem 3.3 shows that they span an integrable generalised distribution D Vs 1 on Λ 2 s 1 . In view of (3.1) the strata of D Vs 1 are the orbits of the action of the connected part of the identity of Aut(g), let us say Aut c (s 1 ), on Λ 2 s 1 . We define M (p) to be a matrix whose entry (i, j) is the j-coefficient of X i at p ∈ Λ 2 s 1 in the basis ∂ x1 | p , . . . , ∂ x6 | p , namely The rank of D Vs 1 at p ∈ Λ 2 s 1 is equal to the rank of M (p). It is simple to calculate that the mCYBE for s 1 is given by the common zeroes of the functions on Λ 2 s 1 of the form It is immediate that the space of solutions to the mCYBE, let us say Y s1 , is not a submanifold in Λ 2 s 1 . Nevertheless, for certain constants λ k ij , with i = 1, . . . , 6 and j, k = 1, 2, 3. Relations (5.4) show that the integral curves of any vector field in V s1 passing through w ∈ Y s1 is contained in Y s1 . In fact, the derivative of the functions f 1 , f 2 , f 3 along an integral curve, Ψ(t), of a vector field X j ∈ V s1 such that Ψ(0) ∈ Y s1 reads λ k ji f k (Ψ(t)), i = 1, 2, 3.
Hence, the values of the f i (Ψ(t)), with i = 1, 2, 3, can be understood as the solutions to a linear system of first-order ordinary differential equations in normal form with constant coefficients and zero initial condition since f i (Ψ(0)) = 0 for i = 1, 2, 3. Hence, f 1 , f 2 , f 3 vanish along Ψ(t) and, since (3.1) shows that the integral curves for all the vector fields in V connect all the points within the same strata of D Vs 1 , one obtains that the functions f 1 , f 2 , f 3 are zero on any strata of D Vs 1 containing a point where f 1 , f 2 , f 3 vanish. It is worth noting that f 1 , f 2 , f 3 are not constants of motion common to all the vector fields of V s1 . We hereafter call f 1 , f 2 , f 3 a Darboux family for the Lie algebra of vector fields X 1 , . . . , X 6 on Λ 2 s 1 . More generally, we propose the following definition and we extend previous results to a more general realm.  It is worth stressing that we require the functions f 1 , . . . , f s and h i jX , for i, j = 1, . . . , s and X ∈ V , in the above definition to be smooth. If all the functions h i jX are equal to zero for every X ∈ V and i = 1, . . . , s, then f j becomes a constant of motion for the vector fields of V on M . If the h i jX are constants for every X ∈ V and i, j = 1, . . . , s, then we say that the Darboux family is linear. In this case, V gives rise to a Lie algebra representation ρ : X ∈ V → D X ∈ End(A), where D X stands for the action of the vector field X on the space of functions A.
The vector fields of V span a generalised distribution, which is integrable and leads to a stratification of M by some disjoint immersed submanifolds which may have different dimensions. Darboux families are interesting to us because the integral curves of vector fields in V passing through the set S A of common zeroes for the elements of A remain within it. Let us prove and analyse this fact, which represents a rather simple generalisation of the argument given for s 1 . Consider a point in the locus S A and consider a basis Y 1 , . . . , Y q of the Lie algebra V of vector fields. Consider an integral curve Ψ(t) of a vector field of V passing through a point Ψ(0) ∈ S A . Then, the time derivative of the functions f 1 (Ψ(t)), . . . , f s (Ψ(t)) is Hence, the values of the f i (Ψ(t)) can be understood as the solutions to the linear system of first-order ordinary differential equations in normal form with t-dependent coefficients with initial condition u 1 = . . . = u s = 0. By the existence and uniqueness theorem, the solution to the previous Cauchy problem is u 1 (t) = . . . = u s (t) = 0. Therefore, the functions f 1 , . . . , f s vanish on the integral curves of X. From this and the decomposition (3.1), we can infer that the functions f 1 , . . . , f s vanish on the strata of the distribution D V containing a point within S A . In other words, the strata of D V containing some point of S A are fully contained in S A . In view of the above, Darboux families can be used to reduce the determination of the strata in M of the generalised distribution D V to determining the strata within S A and out of S A . This will be specially interesting to obtain the strata of D V at points where the generalised distribution is not regular and it may not exist a constant of motion common to all the vector fields in V that can be used to obtain the strata of D V . Note also that if f 1 , . . . , f s form a Darboux family for V , then the set of common zeroes of f 1 , . . . , f s must be the sum (as subsets of M ) of a collection of strata of the generalised distribution D V .
Let us study how to use Darboux families to study solutions to mCYBEs and CYBEs. Let us start by a general result.
g be a Darboux family for V (3) g on Λ 3 g. Then, the space of functions g is a linear Darboux family, then A is also a linear Darboux family.
Proof. Since each X (2) ∈ V g is a fundamental field of the action of Aut(g) on Λ 2 g, its flow is given by the oneparametric group of diffeomorphisms Λ 2 T t induced by a certain one-parametric group of Lie algebra automorphisms T t ∈ Aut(g), t ∈ R. Every g ∈ A is of the form g(r) = f ([r, r]) for some f ∈ A (3) g and every r ∈ Λ 2 g. Then, where f 1 , . . . , f s form a basis of the Darboux family A g , the functions h 1 , . . . , h s are the cofactors of f relative to X (3) and the basis {f 1 , . . . , f s }, and X (3) is the fundamental vector field of the action of Aut(g) on Λ 3 g induced by the one-parameter group {T t } t∈R of Lie algebra automorphisms of g.
g is a linear Darboux family, then h 1 , . . . , h s are constants. Therefore, the cofactors h 1 ([r, r]), . . . , h s ([r, r]) are also constants, which makes A into a linear Darboux family for V g on Λ 2 g.
In short, the following proposition shows that the solutions to the mCYBE on g can be considered as the locus of a Darboux family relative to V g . A similar result can be applied to the CYBE on any Lie algebra g.
• be the annihilator of (Λ 3 g) g , i.e. the subspace of elements of (Λ 3 g) * vanishing on span a linear Darboux family, A, for the Lie algebra V g of fundamental vector fields of the action of Aut(g) on Λ 2 g. Its locus is the set of solutions to the mCYBE for g. Moreover, the components of the CYBE in a coordinate system given by a basis of Λ 2 g * span also a linear Darboux family for V g on Λ 2 g.
Proof. Let us prove that the annihilator of (Λ 3 g) g is a Darboux family for V g , then its flow is given by a one-parameter group of diffeomorphisms of the form Λ 3 T t for certain Lie algebra automorphisms T t ∈ Aut(g) with t ∈ R. Consider a function f ∈ Λ 3 g * such that f (w) = 0 for every w ∈ (Λ 3 g) g , i.e. f ∈ ((Λ 3 g) g ) • . Then, where we have used that (Λ 3 g) g is closed under the extension to Λ 3 g of Lie algebra automorphisms of g and thus is a linear vector field on Λ 3 g and f is a linear function on Λ 3 g, one obtains that X (3) f is a linear function on Λ 3 g vanishing on (Λ 3 g) g . Hence, X (3) f is a linear combination of elements of a basis of ((Λ 3 g) g ) • and ((Λ 3 g) g ) • becomes a linear Darboux family of V (3) g on Λ 3 g. By Theorem 5.2, the space of functions (5.6) becomes a linear Darboux family for V g on Λ 2 g. The locus of (5.6) is the space of solutions of the mCYBE for g.
Since every is linear vector field on a linear coordinate system on Λ 3 g, the space Λ 3 g * is also a Darboux family for V (3) g on Λ 3 g. By Theorem 5.2, one has that the induced by (5.5) family of functions on Λ 2 g is a linear Darboux family relative to V g on Λ 2 g. The Darboux family is indeed spanned by the components of the CYBE of g, i.e. [r, r] = 0, and its locus is its set of solutions.
A natural question is how to determine Darboux families. Here, we provide several results about them that will be useful to as so as to derive new Darboux families from known ones.
Proposition 5.2. If A 1 and A 2 are Darboux families for the same Vessiot-Guldberg Lie algebra V on a manifold M , then the sum A 1 + A 2 is a Darboux family for V on M as well.
The proof of the previous result is immediate. Obviously, the locus of A 1 +A 2 is contained in the locus of A 1 and A 2 . This will be interesting in next sections to distinguish between different strata of the generalised distribution spanned by the vector fields of V g on Y g .
In next sections, it will be relevant to study the linear Darboux families for a certain V g given by a onedimensional A ⊂ (Λ 2 g) * . This is due to the fact that, as seen in Section 7, the sum of such Darboux families as vector spaces will generate new useful Darboux families. We will call such a linear one-dimensional Darboux family a brick. Bricks are one-dimensional representations of V g on Λ 2 g * .
Note that, for every linear Darboux family for V g , bricks are easy to obtain since, in view of (2.3), they are given by the intersection of an eigenvector space for each endomorphism of the form Λ 2 d ∈ gl(Λ 2 g) with d ∈ der(g).
Let us give another interesting proposition, which is an immediate consequence of the fact that the vector fields of V g are linear relative to a linear coordinate set on Λ 2 g. Proposition 5.3. The space Λ 2 g * is a linear Darboux family of functions on Λ 2 g relative to the Lie algebra V g .
Since Λ 2 g * is a linear Darboux family of V g , there exists a linear representation of aut(g) on Λ 2 g * . Its irreducible representations also give rise to Darboux families, which can be summed (as linear spaces) to the Darboux family, or potentially elements thereof, associated with the mCYBE to determine new Darboux families. The locus of such sums will be interested to as when they contain elements of Y g . This will be employed to obtain the strata of V g within Y g and, therefore, inequivalent r-matrices relative to the action of Aut c (g) on Λ 2 g. This in turn will be used to obtain families of inequivalent r-matrices and coboundary cocommutators for real four-dimensional indecomposable Lie algebras [50] in Section 7.

Geometric structure of solutions to mCYBEs and Darboux families
Let us study several details on the geometry of the space of solutions to modified and non-modified classical Yang-Baxter equations, the problem of classification up to Lie algebra automorphisms of coboundary Lie bialgebras, and Darboux families. As previously, we hereafter write V g for the Lie algebra of fundamental vector fields of the action of Aut(g) on Λ 2 g, whilst E g stands for the distribution on Λ 2 g spanned by the vector fields of V g . We recall that we have defined Λ 2 R g := Λ 2 g/(Λ 2 g) g and π g : w ∈ Λ 2 g → [w] ∈ Λ 2 R g stands for the canonical projection onto the quotient space Λ 2 R g. As previously, we hereafter write Y g for the space of r-matrices of g. The action of Aut(g) on Λ 2 g induces an action of Aut(g) on Λ 2 R g of the form (see [35,Lemma 8.3] for details) The above result leads to the following consequence Recall that two actions of a Lie group G, let us say Φ i : ) for every g ∈ G and x ∈ N 1 . It can be proved that if Φ 1 and Φ 2 are equivariant, the Lie algebra of fundamental vector fields of Φ 1 projects, via φ * , onto the Lie algebra of fundamental vector fields of Φ 2 . Moreover, the orbits of Φ 1 project, via φ, onto the orbits of Φ 2 (see [1] for details).
Expression (6.1) shows that the actions of Aut(g) on Λ 2 g and Λ 2 R g are equivariant relative to π g : Λ 2 g → Λ 2 R g. Then, the vector fields of V g project via π g * onto the Lie algebra of fundamental vector fields of the action of Aut(g) on Λ 2 R g. We write E R g for the generalised distribution spanned by the fundamental vector fields of the action of Aut(g) on Λ 2 R g. This means that the strata of the generalised distribution E g project onto the strata of the generalised distribution E R g . As the elements of Aut(g) map solutions to the mCYBE of g onto solutions of the same equation, the orbits of Aut(g) containing a solution to the mCYBE consist of solutions to the mCYBE. Similarly, the orbits of the action of Aut(g) on Λ 2 g containing a solution to the CYBE consist of solutions to the CYBE. Two r-matrices are equivalent up to Lie algebra automorphisms of g if and only if they belong to the same orbit of Aut(g) within Y g .
Recall that Aut(g)/Aut c (g), where Aut c (g) is the connected component of the identity of the Lie group Aut(g), is discrete and therefore countable [13]. In other words, the Lie group Aut(g) is a numerable sum (as subsets) of disjoint and connected subsets of Aut(g) diffeomorphic to Aut c (g). The strata of E g coincide with the orbits of Aut c (g). Therefore, the orbits of Aut(g) are given by a numerable sum of strata of E g . Moreover, each particular orbit of Aut(g) is an immersed submanifold in Λ 2 g of a fixed dimension. Hence, each one of the orbits of Aut c (g), whose sum gives rise to an orbit of Aut(g), must have the same dimension. Similarly, the orbits of Aut(g) on Λ 2 R g are given by a numerable sum (as subsets) of strata of E R g of the same dimension, which are orbits of the action of Aut c (g) on Λ 2 R g. It can be proved that two coboundary cocommutators can be equivalent up to Lie algebra automorphisms even when their associated r-matrices are not. For instance, the zero cocommutator δ 0 : g → g ∧ g can take the form 0 = δ 0 = [r, ·] for r = 0 or for any other r 1 ∈ (Λ 2 g) g \{0}. Nevertheless, the 0 and r 1 cannot be connected by the action of an element of Aut(g) on Λ 2 g, since the action of Aut(g) on Λ 2 g is linear and 0 ∈ Λ 2 g is an orbit. In spite of that, the cocommutators generated by 0 and r 1 are the same. Let us give the conditions ensuring that two coboundary cocommutators on g are equivalent up to a Lie algebra automorphism of g. Proposition 6.1. Two coboundary cocommutators δ i : v ∈ g → [r i , v] ∈ g ∧ g, with i = 1, 2 and r i ∈ Y g , are equivalent under a Lie algebra automorphism of g if and only if there exists T ∈ Aut(g) such that (Λ 2 T )(r 1 ) and r 2 belong to the same equivalence class in Λ 2 R g. Proof. It is clear that the transformation of δ 1 by T reads, by the properties of the Schouten bracket, as follows Hence, and the last condition amounts to the fact that Λ 2 T r 1 − r 2 ∈ (Λ 2 g) g , which implies that Λ 2 T r 1 , r 2 belong to the same equivalence class in Λ 2 R g. Note that two r-matrices are equivalent (relative to the action of an element of Aut(g) on Λ 2 g) if and only if they belong to the same family of strata of the distribution E g giving rise to an orbit of Aut(g) in Y g . Nevertheless, two r-matrices give rise to two equivalent coboundary cocommutators if and only if they belong to the same orbit of Aut(g) on Λ 2 R g. In other words, we have proven the following proposition. Proposition 6.2. There exists a one-to one correspondence between the r-matrices for g that are equivalent up to an element of Aut c (g) and the strata of the generalised distribution E g within Y g . Moreover, there exists a one-to-one correspondence between the families of equivalent (up to Lie algebra automorphisms of g) coboundary cocommutators of a Lie algebra g and the orbits of the action of Aut(g) on π g (Y g ) ⊂ Λ 2 R g. Every such an orbit in π g (Y g ) is the sum (as subsets) of a numerable collection of strata of the same dimension of the generalised distribution E R g . In practice, we shall obtain each orbit of Aut(g) on Y g as a numerable family of strata of E g in Y g . Such orbits represent the families of r-matrices that are equivalent up to an element of Aut(g). Then, we will derive all strata in Y g that map onto the same space in Λ 2 R g. This last task will give us, along with the orbits of Aut(g) on Y g , the equivalent classes of coboundary cocommutators on g up to Lie algebra automorphisms of g.
Let us study how Darboux families can be employed to obtain the orbits of Aut(g) on Λ 2 g and Λ 2 R g. In particular, Darboux families are employed here to identify the strata of E g . Proposition 6.3. The locus, ℓ g , of a Darboux family A E g relative to the Lie algebra V g on Λ 2 g is a sum (as subsets) of the orbits of the action of Aut c (g) on Λ 2 g containing a point in ℓ g . If ℓ g is a connected submanifold in Λ 2 g of dimension given by the rank of the generalised distribution E g , then ℓ g is a strata of the generalised distribution E g .
Proof. For each point p in ℓ g the orbit O p of Aut c (g) passing through p has a tangent space spanned by the vector fields of V g . By the definition of a Darboux family for V g and the fact that the functions of A E g vanish at p, one has that all functions of A E g are equal to zero on O p (see Section 5). Hence, O p is contained in ℓ g . Then, ℓ g is the sum (as subsets) of orbits of Aut c (g) passing through the points of ℓ g .
Recall that the strata of E g are the orbits of Aut c (g) acting on Λ 2 g. Hence, ℓ g is a sum of strata of E g . If the rank of E g on ℓ g is equal to dim ℓ g , where ℓ g is assumed to be a connected submanifold, then ℓ g is locally generated around any point p ∈ ℓ g by the action of Aut c (g) on that point. Since ℓ 1 is connected, it is wholly generated by the action of Aut c (g) and it becomes a strata of E g .
Note that we are interested only in those loci of Darboux families of V g contained in the space Y g of solutions to the mCYBE of g.
It is interesting that necessary conditions for the existence of a Lie algebra automorphism connecting two rmatrices can be given using Λ 3 g. Consider, for instance, the following proposition, whose proof is straightforward. Proposition 6.4. If r 1 and r 2 are two equivalent r-matrices for a Lie algebra g, then, [r 1 , r 1 ] and [r 2 , r 2 ] belong to the same orbit of Aut(g) acting on (Λ 3 g) g .
The necessary condition in Proposition 6.4 is not sufficient. For instance, it may happen that two r-matrices are solutions to the classical Yang-Baxter equation without being equivalent (as seen in Table 2). From a practical point of view, Proposition 6.4 shows that the determination of the orbits of the action of Aut(g) on Λ 3 g may be of interest. Following the reasoning for studying the equivalence of r-matrices in Λ 2 g, the orbits of the action of Aut(g) on Λ 3 g can be obtained by obtaining the strata of the distribution, E in an analogous way to the method employed to study E g . Then, the action of one element of each connected component of Aut(g) on the strata of V Proposition 6.5. If two r-matrices r 1 , r 2 are equivalent, then the rank of r 1 , r 2 , as bilinear mappings on g * , are the same.
Proof. Recall that r-matrices can be considered as the antisymmetric bilinear mappings on g * associated with them. The action of Aut(g) on an r-matrix does not change its rank as a bilinear antisymmetric mapping. Hence, equivalent r-matrices must have the same rank.
Although r-matrices whose bilinear antisymmetric mappings on g * have different rank are not equivalent up to Lie algebra automorphisms of g, it may happen that r-matrices related to the same rank are not equivalent. In fact, although there always exists in this latter case an A ∈ GL(g) acting on Λ 2 g mapping one element of this space into another of the same order (A can be obtained by mapping both bivectors into a canonical form), A does not necessarily belong to Aut(g). Hence, both bivectors need not be equivalent up to Lie algebra automorphisms of g. Many examples of this will appear in Table 2 We hereafter assume g to be a real four-dimensional and indecomposable Lie algebra. For the sake of completeness, the structure constants for every g in a basis {e 1 , e 2 , e 3 , e 4 } are given in Table 1. We hereafter endow Λ 2 g with a basis {e 12 := e 1 ∧ e 2 , e 13 := e 1 ∧ e 3 , e 14 := e 1 ∧ e 4 , e 23 := e 2 ∧ e 3 , e 24 := e 2 ∧ e 4 , e 34 := e 3 ∧ e 4 }, while its dual basis is denoted by {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 }. Additionally, Λ 3 g is endowed with a basis {e 123 , e 124 , e 134 , e 234 }. Moreover, due to the bilinearity and symmetry of the Schouten bracket on Λ 2 g, it is enough to know [e ij , e kl ], with 1 ≤ i < k ≤ 4 or i = k ∈ {1, . . . , 4}, 1 ≤ j ≤ l ≤ 4, to determine the Schouten bracket on the whole Λ 2 g. In Tables  3-5, we summarise several relevant Schouten brackets between the basis elements of g, Λ 2 g, and Λ 3 g that can easily be obtained via Table 1 and, eventually, by means of a symbolic computation program. Anyway, we add such calculations to easily follow the proofs of our following results. The classes of Lie algebras in Table 1 may contain several non-isomorphic Lie algebras depending on several parameters, e.g. α, β. To simplify the notation, we will not detail specific values of the parameters when we refer to properties of a whole class of Lie algebras and/or it is clear the particular case we are discussing from the context.
It is interesting to discuss some physical and mathematical applications of the Lie bialgebras given in our list and their relations to some previous works. For instance, the nilpotent Lie algebra n 1 corresponds to the so-called Galilean Lie algebra, which is deformed in the study of XYZ models with the quantum Galilei group appearing in [12]. Lie bialgebras defined on Lie algebras of the type n 1 also occur in [42,43]. The coboundary Lie bialgebras on the Lie algebra s 6 , which corresponds to the so-called harmonic oscillator algebra, are obtained in [6] without studying the equivalence up to Lie algebra automorphisms. In that work, the quantisations and R-matrices of such a sort of Lie bialgebras are also studied. In particular, equation (3.7) in [6] matches the mCYBE obtained in Section 7.6. The work [7] proves that all Lie bialgebras on the harmonic oscillator algebra are coboundary ones. This shows that our classification of Lie bialgebras on s 6 finishes the classification of all Lie bialgebras on the harmonic oscillator algebra. On the other hand, the Lie algebra s 8 with α = −1/2 is isomorphic to the Lie algebra AḠ 2 (1) in [42]. The Lie algebra s 7 provides a central extension of Caley-Klein Lie algebras analysed in [10, eq. (3.13)]. Other cases of real four-dimensional indecomposable Lie bialgebras can be found in [4,44], where, for instance, two-dimensional central extended Galilei algebras are studied.

Lie algebra s 1
The structure constants for Lie algebra s 1 are given in Table 1, while relevant Schouten brackets between the elements of the bases of s 1 , Λ 2 s 1 , Λ 3 s 1 to be used hereafter are displayed in Tables 3-5. In particular, one sees from such tables that (Λ 2 s 1 ) s1 = e 12 and (Λ 3 s 1 ) s1 = 0.  Table 1: Structure constants for real four-dimensional indecomposable Lie algebras according to the WŠ classification. Such Lie algebras are denoted by s 4,k , n 4,1 for 1 ≤ k ≤ 12 in the WŠ classification. Nevertheless, the subindex concerning the dimension of the Lie algebra will be removed in our work to simplify the notation. The letters n and s are used to denote nilpotent and solvable (but not nilpotent) Lie algebras, respectively. To avoid duplicities of Lie algebras, the parameters α, β of s 3 must satisfy additional restrictions that were not detailed explicitly in [50] (see [50, p. 228]). Such restrictions will be determined in Subsection 7.3.
To obtain the classification of r-matrices for s 1 up to Lie algebra automorphisms thereof, we will rely on the determination of the strata of the distribution E s1 (see Proposition 6.2) via Darboux families for V s1 and a few elements of Aut(s 1 ).
By Proposition 6.3, the locus of a Darboux family for a Lie algebra V s1 is the sum (as subsets) of strata of the generalised distribution E s1 spanned by the vector fields of V s1 . In particular, we are interested in those loci being a submanifold within Y s1 of dimension matching the rank of E s1 on them. This is due to the fact that, in view of Proposition 6.3, their connected components are the orbits of Aut c (s 1 ) on Y s1 . With this aim, we derive the fundamental vector fields of the action of Aut(s 1 ) on Λ 2 s 1 . By Remark 2.1, the derivations of s 1 take the form (5.1), and a basis of V s1 is given by (5.2).
For an element r ∈ Λ 2 s 1 , we get This expression can easily be derived by using the properties of the algebraic Schouten bracket and the structure constants for s 1 . A similar computation would not be difficult even for many higher-dimensional Lie algebras. Since (Λ 3 s 1 ) s1 = 0, the mCYBE and the CYBE for s 1 are equal and read It is interesting that, since (Λ 3 s 2 ) s1 = 0, every r-matrix for s 1 amounts to a left-invariant Poisson bivector on s * 1 (cf. [15]). Let us apply to s 1 the following procedure to obtain the orbits of Aut c (g) on Y g for a Lie algebra g of arbitrary dimension, i.e. the strata of E g within Y g . We start by considering a locus ℓ 1 of a Darboux family of V g on Λ 2 g. As we are interested in determining the strata of E g within Y g , we look for an ℓ 1 containing some solution to the mCYBE, namely ℓ 1 ∩ Y g = ∅. Recall that the strata of E g are completely contained within the locus or completely contained off the locus of every Darboux family for V g . This divides the strata of E g on Y g into those within or outside ℓ 1 . If ℓ 1 ∩ Y g is a submanifold of dimension given by the rank of E g on it, its connected parts are the strata of E g in ℓ 1 ∩ Y g . Otherwise, to obtain the strata within ℓ 1 ∩ Y g , we look for a new Darboux family for V g containing the previous Darboux family so that its new locus, let us say ℓ 2 , will satisfy ℓ 2 ∩ Y g = ∅. This process is repeated until we obtain a locus, ℓ k , that is a submanifold of dimension equal to the rank of E g on it. In other words, the connected components of the locus ℓ k are the orbits of Aut c (g) in ℓ k .
To determine all the strata of E g on Y g ∩ (ℓ k ) c , where (ℓ k ) c stands for the complementary subset of ℓ k in Λ 2 g, which is always open, one considers a subspace of the final Darboux family and the previous procedure is applied again iteratively to the obtained Darboux family on Λ 2 g\ℓ k . At the end, we obtain another family of strata of E g and the procedure can be repeated to search for remaining strata of E g in Y g . The above-described process, for all four-dimensional indecomposable Lie algebras, allows us to obtain the orbits of Aut c (g) on Y g .
To represent schematically our procedure, we use a diagram, which is hereafter called a Darboux tree. Every Darboux family of the above-mentioned method is described by a set of boxes going from an edge of the diagram on the left to one of the edges on the right. The squared boxes of the type f = 0, for a certain function f , give the generating functions of the Darboux family while the squared boxes of the type f = 0, give the conditions restricting the manifold in Λ 2 g where the Darboux family and V g are restricted to. The connected parts of the loci of all the Darboux families represented in a Darboux tree give rise to a decomposition of Y g into orbits of Aut c (g). To help understanding the calculations, some oval boxes with additional information, e.g. explaining why other possibilities are not considered, are given. It is worth noting that bricks where employed to generate Darboux families. In particular, the Darboux tree of s 1 is given below. In this case, it is immediate that x 5 and x 6 are bricks for V s1 . We use this fact to generate Darboux families of the following Darboux tree. The connected components of the loci of the Darboux families of the previous Darboux tree give the orbits of Aut c (s 1 ) on Y s1 . Results are given in Table 2. Let us study the equivalence of r-matrices up to action of the whole Aut(s 1 ). To do so, we use that the group of automorphisms of s 1 reads This Lie group is easy to be derived from the structure constants of s 1 . In reality, it is enough for our purposes to consider an element of each connected component of Aut(s 1 ). It is remarkable that Lie algebra automorphism groups can rather easily be obtained for Lie algebras of relatively high dimension and/or satisfying special properties, e.g. for semisimple Lie algebras [40]. In our case, it is immediate that Aut(s 1 ) has four such components. One element of each connected component of Aut(s 1 ) and its extension to Λ 2 s 1 are given by The orbits of Aut(s 1 ) in Y s1 are given by the action of Λ 2 T λ1,λ2 on the orbits of Aut c (s 1 ) on Y s1 . By using all mappings Λ 2 T λ1,λ2 , we can verify whether some of the orbits of Aut c (s 1 ) on Y s1 can be connected among themselves by a Lie algebra automorphism of s 1 . Our results are summarised in Table 2.
Recall that (Λ 2 s 1 ) s1 = e 12 . Consequently, all orbits of Aut(s 1 ) on Y s1 mapping onto the same space in Λ 2 R s 1 via π s1 give equivalent coboundary coproducts up to the action of elements of Aut(s 1 ). In particular, we get five classes of coboundary coproducts induced by the following classes of r-matrices: a) I ± , b) II, III ± , c) IV, V ± , d) VI, and e) VII, VIII ± . We recall that all given r-matrices are solutions to the CYBE.

Lie algebra s 2
Let us apply the formalism given in the previous section to Lie algebra s 2 . Lie algebra s 2 has structure constants given in Table 1. Relevant Schouten brackets between the bases of elements of s 2 , Λ 2 s 2 , and Λ 3 s 2 are given in Tables 3-5. Remarkably, (Λ 2 s 2 ) s2 = 0 and (Λ 3 s 2 ) s2 = 0.
By Remark 2.1, it follows that which gives rise, by lifting these derivations to Λ 2 s 2 , to the basis of V s2 of the form Meanwhile, for r ∈ Λ 2 s 2 , one has Since (Λ 3 s 2 ) s2 = 0, the mCYBE and the CYBE are the same and read With the previous information, we are ready to obtain the classification of orbits of the action of Aut c (s 2 ) on the subset of r-matrices Y s2 ⊂ Λ 2 s 2 by using Darboux families. We shall start by using bricks. The only brick of s 2 is x 6 . The procedure is accomplished as in the previous section and it is summarised in the following Darboux tree. The nine connected parts of the submanifolds 0, I ± , II ± , III, IV ± are the orbits of Aut c (s 2 ) on Y s2 . It is simple to see which of them are further connected through an element of Aut(s 2 ). In result, the orbits of Aut(s 2 ) on Y s2 are given by the eight submanifolds 0, I − , I + , II − , II + , III, IV − , IV + given in Table 2. Since (Λ 2 s 2 ) s2 = 0, each such a submanifold gives a family of equivalent coboundary coproducts that is non-equivalent to r-matrices within remaining submanifolds. This gives all possible classes of non-equivalent coboundary coproducts. As in the previous subsection, all r-matrices are solutions to the CYBE in s 2 .

Lie algebra s 3
The structure constants of the Lie algebras of type s 3 are given in Table 1. It was noted in [50, p. 228] that additional restrictions must be imposed on the parameters α, β given in Table 1 to avoid the repetition of isomorphic Lie algebras within the Lie algebra class s 3 . Such restrictions were not explicitly detailed in [50], but it is immediate that isomorphic cases within Lie algebras of the class s 3 can be classified by the adjoint action of e 4 on e 1 , e 2 , e 3 . In particular, if |α| = |β|, then the Lie algebras with parameters (α, β) and (β, α) are isomorphic relative to the Lie algebra isomorphism that interchanges e 2 with e 3 and leaves e 1 invariant. It will be relevant in what follows that the lift of this Lie algebra automorphism to Λ 2 s 3 interchanges x 1 with x 2 , x 5 with x 6 , it maps x 4 to −x 4 , and it leaves x 3 invariant. Due to the previous isomorphism, we restrict ourselves to the case α ≥ β when |α| = |β|.
Using the structure constants given in Table 1 and the induced Schouten brackets between the relevant to our purposes basis elements of s 3 , Λ 2 s 3 , and Λ 3 s 3 in Tables 3-5 Since the Lie algebra class s 3 contains so many subcases that relevant properties may change from one to another subcase with given parameters (α, β), e.g. the Lie algebra automorphism group, we will develop here a modification of our method consisting of applying the Darboux family method to the fundamental vector fields of the action of the Lie group Aut all (s 3 ) of all common Lie algebra automorphisms for all parameters α, β and analysing its relation to the Lie algebra automorphisms for each case, s α,β 3 , to obtain our final classification. To simplify the notation, we skip the parameters α, β in s α,β 3 when they are not needed to understand what we are talking out, e.g. if we jest care about the dimension of the linear space Λ 2 s 3 .
By Remark 2.1, one obtains the following common Lie algebra of derivations of s 3 for all the values of α and β: and, after lifting the elements of a basis of der all (s 3 ) to Λ 2 s 3 , we obtain a basis of V all s3 , namely the Lie algebra of fundamental vector fields of the action of the Lie algebra automorphism group Aut all (s 3 ) common to all α, β acting on Λ 2 s 3 , of the form It is worth stressing that if α = β and/or one on the coefficients α, β are equal to one, then the Lie algebra of derivations of the particular s α,β 3 is larger due to the existence of Lie algebra automorphisms of s α,β 3 leaving invariant the eigenspaces of ad e4 acting on e 1 , e 2 , e 3 (more detailed calculations can be seen at the end of this subsection). By dealing with der all (s 3 ), we shall derive the orbits of the connected part of the identity of the group of common Lie algebra automorphisms for all α, β, i.e. Aut all (s 3 ), on each Y s α,β 3 via Darboux families. To obtain the equivalence of r-matrices up to the action of Aut(s α,β 3 ) for each pair (α, β), we will use the action of elements of Aut(s α,β 3 ) not contained in Aut all,c (s 3 ) for each particular pair of parameters (α, β).

Case d):
In this case, ad e4 acts on e 1 , e 2 , e 3 having three different eigenvalues and this leads to the fact that the only derivations are those common to all α, β, i.e. der(s α,β 3 ) = der all (s 3 ). The connected parts of the loci of the Darboux families of the above Darboux tree can be found in Table 2 As in previous sections, we only need for our purposes one element of each connected part of Aut(s α,β 3 ), which has eight ones. An element of each connected component and its lift to Λ 2 s 3 are given by  Table 2. To obtain the equivalence classes of coboundary coproducts for each α, β, it is enough to identify the orbits in Table 2 whose elements are the same up to an element of (Λ 2 s α,β 3 ) s α,β

3
. In particular, we have the following subcases: To obtain the orbits of Aut(s α,α 3 ) on Y s α,α 3 from the orbits of Aut all,c (s 3 ), it is necessary to write Aut(s α,α 3 ) as a composition of Aut all,c (s 3 ) with certain Lie algebra automorphisms of s α,α 3 so that their composition generates Aut(s α,α 3 ). This can be done by using the Lie algebra automorphisms of s α,α 3 of the form T A := Id ⊗ A⊗Id, for A ∈ GL(2, R). Then, Λ 2 T A = A ⊗ Id ⊗ (det A)Id ⊗ A. By taking the action of these Λ 2 T A on the strata of the distribution spanned by V all s3 in Y s α,α

3
. Our results are summarised in Table 2.
To obtain the equivalence classes of coboundary coproducts for each α = β, it is enough to identify the orbits in Table 2 whose elements are the same up to an element of (Λ 2 s α,α 3 ) s α,α 3 = 0. Therefore, the result is given by the classes of equivalent r-matrices detailed in Table 2.
Case b): This time α = 1 = β. Therefore, To derive the orbits of Aut given by T A := A⊗Id ⊗ Id, for A ∈ GL(2, R). Then . Our results are summarised in Table 2. To obtain the equivalence classes of coboundary coproducts for each β, it is enough to identify the equivalence classes of r-matrices in Table 2 Due to the larger family of symmetries for α = β = 1, the Lie algebra V s 1,1 3 is spanned by The Darboux tree is very simple and becomes It is immediate that the above gives rise to two non-zero coboundaries given in Table 2. Since in this case (Λ 2 s 1,1 3 ) s 1,1 3 = 0, each class of equivalent r-matrices amounts to a class of equivalent coboundary coproducts.
As the Lie algebras of the class s 4 depend on a parameter α ∈ R\{0}, the space of derivations depend on α. It is indeed the same for all values of α ∈ R\{0, 1}, and it becomes larger for α = 1. Due to this, We shall proceed as in Subsection 7.3. By Remark 2.1, the space of derivations of s 4 for all possible values of α read der all (s 4 ) := By extending the previous derivations to Λ 2 s 4 and using Remark 2.1, we obtain a basis of V all s4 of the form We recall that these vector fields span the Lie algebra of fundamental vector fields of the action on Λ 2 s 4 of the Lie algebra automorphisms Aut all (s 4 ) that are common for all values of α ∈ R\{0}.
For an element r ∈ Λ 2 s 4 , we get If α = −2, we have (Λ 3 s 4 ) s4 = 0. Thus, the mCYBE and the CYBE are equal in this case and they read The connected parts of the subspaces denoted in the above diagram are the orbits of Aut all,c (s 4 ) on Y s4 . To obtain the orbits of Aut all (s 4 ) on Y s4 , where we verify whether some of the connected components of the orbits of Aut all,c (s 4 ) on Y s4 are additionally connected by an element of Aut all (s 4 ). To do so, we use, as previously, the lift to Λ 2 s 4 of one element of each connected component of Aut all (s 4 ), for instance For α ∈ R\{1, 0}, one has that Aut(s α 4 ) = Aut all (s 4 ), where s α 4 stands for the Lie algebra s 4 for a fixed value of α. Then, the classes of equivalent r-matrices (up to Lie algebra automorphisms of s 1 4 ) on the Lie algebra s α 4 can easily be obtained and they are summarised in Table 2 Note that, for values |α| = 1, not all the above classes may be simultaneously available as they arise for particular values of α.
Let us consider now the case of the Lie algebra s 1 4 , i.e. the Lie algebra of the class s 4 for α = 1. In this case, the group of Lie algebra automorphisms is slightly larger than for remaining admissible values of α. In particular, for every λ, µ ∈ R. This will give the final orbits detailed in Table 2. Since (Λ 2 s 1 4 ) s 1 4 = 0, the families of equivalent coboundary coproducts are given by the ones induced by the families of equivalent r-matrices.
By Remark 2.1, we obtain that the derivations of s 5 read der(s 5 ) := which give rise to the basis of V s5 of the form For an element r ∈ Λ 2 s 5 , we get If α + 2β = 0, then (Λ 3 s 5 ) s5 = 0. Thus, the mCYBE and the CYBE are equal in this case and they read For α + 2β = 0, we have (Λ 3 s 5 ) s5 = e 123 and the mCYBE reads Since x 3 is the only brick for s 5 , we start our Darboux tree with the cases x 3 = 0 and x 3 = 0. The full Darboux tree is presented below.
Orbits of Aut c (s 5 ) are given by the connected parts of the loci of the above Darboux tree. Solutions are described in Table 2.
The automorphism group of s 5 reads Since the subgroup of GL(2, R) of matrices of the form can be parametrised via φ ∈ [0, 2π[ and µ := [(T 2 2 ) 2 + (T 3 2 ) 2 ] 1/2 ∈ R + by setting T 2 2 = µ cos φ and T 3 2 = µ sin φ, there are two connected components of Aut(s 5 ). A representative of each connected part and its lift to Λ 2 s 5 read (7.5) Using techniques from previous sections, we can easily verify whether the orbits of Aut c (s 5 ) are additionally connected by a Lie algebra automorphism of s 5 via Λ 2 T λ . Our results are summarised in Table 2. Moreover, for β = 0, the classes of coboundary coproducts are induced from the following families of r-matrices: Meanwhile, for β = 0, the list of families of equivalents coboundary coproducts read a) II ± , b) I, III ± , c) IV, V β=0 ± .

Lie algebra s 6
Structure constants of Lie algebra s 6 are given in Table 1. Using this information, we can compute the Schouten brackets of elements of s 6 , Λ 2 s 6 , and Λ 3 s 6 (see Tables 3-5 which gives rise to the basis of V s6 of the form For an element r ∈ Λ 2 s 6 , we get Since (Λ 3 s 6 ) s6 = e 123 , the mCYBE reads whereas the CYBE is Note that the CYBE obtained above is exactly the result obtained in [6, eq. (3.7)] under the substitution x 5 = −α + , x 6 = −α − , x 4 = ξ and x 3 = ϑ. Since x 5 , x 6 are the bricks of s 6 , our Darboux tree starts with cases x i = 0 and x i = 0, for i ∈ {5, 6}. The full Darboux tree is presented below.
The connected parts of the subsets denoted in the above Darboux tree are the orbits of Aut c (s 6 ). The Lie algebra automorphism group of s 6 reads Aut(s 6 ) = One element of each connected component of Aut(s 6 ), which are eight, and their lifts to Λ 2 s 6 are given by 6) where θ(x) stands for the Heaviside function and λ 1 , λ 2 ∈ {1, −1}. By using the previous information, we can verify whether some of the orbits of the action of Aut c (s 6 ) on Y s6 can be connected by an element of Aut(s 6 ), which gives rise to the classes of equivalent r-matrices up to the action of Lie algebra automorphisms of s 6 . Since (Λ 2 s 6 ) s6 = 0, each class of equivalent r-matrices gives rise to a separate class of coboundary Lie bialgebras on s 6 . Hence, the classes of equivalent coboundary Lie bialgebras are given by a) I, I ext b) II, c), III, d) IV, e) V, V ext f ) VI, VI ext , g) VII |k| ={0,1} , h) VIII, VIII ext , i) IX, IX ext .

Lie algebra s 7
The structure constants of Lie algebra s 7 are given in Table 1. Using this information, we can compute some relevant Schouten brackets between the elements of the bases of s 7 , Λ 2 s 7 , and Λ 3 s 7 (see . Note that, from these tables, we get (Λ 2 s 7 ) s7 = 0 and (Λ 3 s 7 ) s7 = e 123 .
By Remark 2.1, one obtains that der(s 7 ) := which give rise to the basis of V s7 of the form Then, Since (Λ 3 s 7 ) s7 = e 123 , then the mCYBE reads Meanwhile, the CYBE takes the form The Darboux tree is presented below.
The connected parts of the subsets denoted in the above Darboux tree are the orbits of Aut c (s 7 ) in Y s7 . By using the extension of the automorphism group Aut(s 7 ) to Λ 2 s 7 , we can verify whether some of these parts are additionally connected by a Lie algebra automorphism of s 7 . The results are summarised in Table 2. As before, each family of r-matrices induces a separate class of coboundary coproducts.

Lie algebra s 8
The structure constants for the Lie algebra s 8 are given in Table 1. Recall that α ∈] − 1, 1]\{0}. As in the previous cases, one can obtain some relevant Schouten brackets between the basis elements of s 8 , Λ 2 s 8 , and Λ 3 s 8 (see Tables  3-5). In view of Tables 1,3 Since (Λ 3 s 8 ) s8 = 0, the mCYBE reads Let us now consider two cases given by α ∈] − 1, 1[\{0} and α = 1. In the first case, the space of derivations gives rise to the basis of V s α 8 of the form The following diagram depicts the Darboux families that give us the orbits of Aut c (s α 8 ) on Y s α 8 . Since x 5 , x 6 are the bricks for every s α 8 , our Darboux tree starts with the cases x i = 0 and x i = 0, for i ∈ {5, 6}. The full Darboux tree is presented below.
No solutions The Lie algebra automorphism group of s α 8 for each α ∈] − 1, 1[\{0}, reads In reality, we are only concerned with obtaining one element of Aut(s α 8 ) and its lift to Λ 2 s 8 for each one of its connected components. For instance, we can choose (7.10) As in previous sections, the maps Λ 2 T λ1,λ2 allow us to identify the orbits of the action of Aut(s α 8 ) on Y s α 8 . Our results are presented in Table 2 Let us know study the case α = 1 using the previous results. Since (7.7) for α = 1 is a Lie subalgebra of the space of derivations for s α 8 given in (7.8), the Lie algebra spanned by the vector fields of (7.9) for α = 1 is a Lie subalgebra of the Lie algebra of fundamental vector fields of the action of Aut(s 1 8 ) on Λ 2 s 8 . Hence, the loci of the above Darboux families allow us to characterise the strata of the distribution spanned by the vector fields (7.9) for α = 1, which in turn gives as the orbits of a Lie subgroup of Aut(s 1 8 ). To easily follow our discussion, we detail that To obtain the orbits of Aut(s 1 8 ) on Y s 1 8 , it is enough to write Aut(s 1 8 ) as a composition of the previous subgroup with certain Lie algebra automorphisms of Aut(s 1 8 ), e.g. the Lie algebra automorphisms T A = (det A)Id ⊗ A ⊗ Id for every A ∈ GL(2, R). Hence, The action of the Λ 2 T A on the loci of the Darboux families of (7.9) assuming α = 1 allows us to obtain the orbits of the action of Aut(s 1 8 ) on Y s 1 8 . In particular, one obtains the subsets given in Table 2 7.9 Lie algebra s 9 As previously, we use the structure constants for s 9 in Table 1 to compute the Schouten brackets given in Tables 3-5. Moreover, (Λ 2 s 9 ) s9 = 0 and (Λ 3 s 9 ) s9 = 0 for every α > 0. These calculations are enough to verify the remaining results of present subsection. By Remark 2.1, one obtains that The obtained derivations give rise to the basis of V s α 9 of the form For an element r ∈ Λ 2 s 9 , we get And since (Λ 3 s 9 ) s9 = 0 for every value of α, the mCYBE and the CYBE are equal and read The Darboux tree for the class s 9 is presented below. If we define ∆ := (T 2 2 ) 2 + (T 2 3 ) 2 , the Lie algebra automorphisms group of each s α 9 reads Using ideas from Section 7.5, we obtain that each Aut(s α 9 ) has one connected component. Consequently, the orbits of Aut(s α 9 ) on Λ 2 s α 9 are the strata of E s α 9 . Since (Λ 2 s 9 ) s9 = 0 for every α > 0, the strata of E s α 9 within Y s α 9 amount for the families of equivalent coboundary Lie bialgebras on s α 9 . Our final results are summarised in Table  2. 7.10 Lie algebra s 10 As in the previous cases, we hereafter use the structure constants for Lie algebra s 10 in Table 1  which give rise to the basis of V s10 of the form For an element r ∈ Λ 2 s 10 , we get The connected parts of the subspaces denoted in the Darboux tree are the orbits of Aut c (s 10 ) within Y s10 . To obtain the orbits of Aut(s 10 ) within Y s10 , we proceed as in previous sections and we derive the automorphism group of s 10 , which takes the form In reality, we do not need the full group. It is enough to note that there are two connected components of Aut(s 10 ), represented by two automorphisms that read, along with their extensions to Λ 2 s 10 , as follows By applying Λ 2 T λ to the orbits of Aut c (s 10 ) in Y s10 , depicted in Table 2, we obtain the orbits of Aut(s 10 ) in Y s10 . Since (Λ 2 s 10 ) s10 = 0, such orbits amount to the classification of equivalent coboundary Lie bialgebras up to Lie algebra automorphisms of s 10 .

Lie algebra s 11
As previously, the information in Tables 1, 3 Such derivations give rise to the basis of V s11 of the form For an element r ∈ Λ 2 s 11 , we get Since (Λ 3 s 11 ) s11 = {0}, the mCYBE and the CYBE are equal and read Since x 5 , x 6 are the bricks for s 11 , our Darboux tree starts with the cases x i = 0 and x i = 0, for i ∈ {5, 6}. The Darboux tree is presented below. The above Darboux tree gives the orbits of Aut c (s 11 ) in Y s11 . To obtain the equivalence classes of r-matrices up to Lie algebra automorphisms of Aut(s 11 ), we obtain that In fact, only one representative for each connected component is needed for our purposes. There are four connected components of Aut(s 11 ), each one represented by one of the following automorphisms, which are given together with their lifts to Λ 2 s 11 , namely (7.12) The Λ 2 T λ1,λ2 allow us to check whether the orbits of Aut c (s 11 ) are connected by an element of Aut(s 11 ) acting on Λ 2 s 11 . The equivalence classes of such r-matrices (up to the action of elements of Aut(s 11 ), as standard) are given in Table 2. Since (Λ 2 s 11 ) s11 = 0, each family of r-matrices gives rise to a separate class of equivalent coboundary Lie bialgebras on s 11 .

Lie algebra s 12
The structure constants for Lie algebra s 12 are given in Table 1. This along with the selected Schouten brackets between basis elements of s 12 , Λ 2 s 12 , and Λ 3 s 12 described in Tables 3-5 The previous derivations give rise to the basis of V s12 given by For an element r ∈ Λ 2 s 12 , we get Since (Λ 3 s 12 ) s12 = 0, the mCYBE and CYBE are equal and read Since x 6 is the only brick for s 12 , our Darboux tree starts with the cases x 6 = 0 and x 6 = 0. The full Darboux tree is given next.
No solutions No solutions The connected parts of the loci of the Darboux families of the Darboux tree are the orbits of Aut c (s 12 ) within Y s12 . These are given by the connected parts of the subsets given in Table 2. Let us obtain the orbits of Aut(s 12 ) on Y sl 12 . The Lie algebra automorphism group of s 12 reads There are two connected components of Aut(s 12 ). One such element for each connected component of Aut(s 12 ) and their extensions to Λ 2 s 12 are given by By using Λ 2 T ± on the loci of the above Darboux tree, we obtain the orbits of Aut(s 12 ) that are given by the classes of equivalent r-matrices detailed in Table 2. Since (Λ 2 s 12 ) s12 = 0, each family of equivalent r-matrices gives rise to a separate class of equivalent coboundary Lie bialgebras.

Lie algebra n 1
The structure constants for the Lie algebra n 1 are given in Table 1, while relevant Schouten brackets between the elements of bases of n 1 , Λ 2 n 1 , and Λ 3 n 1 to be used hereafter are displayed in Tables 3-5. From these calculations, one obtains that (Λ 2 n 1 ) n1 = e 12 and (Λ 3 n 1 ) n1 = e 123 , e 124 . By Remark 2.1, one has that the derivations 2 of n 1 take the form which, by lifting them to Λ 2 n 1 , give rise to the basis of V n1 of the form For an element r ∈ Λ 2 n 1 , we get [r, r] = 2(x 4 x 5 − x 2 x 6 )e 123 + 2(x 2 5 − x 3 x 6 )e 124 + 2x 5 x 6 e 134 + 2x 2 6 e 234 .
Since (Λ 3 n 1 ) n1 = e 123 , e 124 , the mCYBE differs from the CYBE and reads The Darboux tree for this Lie algebra reads As in all previous sections, the orbits of Aut c (n 1 ) in Y n1 are given by the connected components of the loci of the Darboux families of the above Darboux tree. Results can be found in Table 2. Let us again obtain the orbits of Aut(n 1 ) in Y n1 . The automorphism group of n 1 takes the form It has four connected components represented by the following Lie algebra automorphisms, which are also lifted to Λ 2 n 1 : The action of the Λ 2 T λ1,λ2 on the loci of the above Darboux tree gives the families of equivalent r-matrices given in Table 2. Since (Λ 2 n 1 ) n1 = e 12 and using the information in Table 2, we see that the families of equivalent coboundary Lie bialgebras on n 1 are given by orbits of Aut(n 1 ) with the exception that the class I and the zero class (given by the zero r-matrix) give the Lie bialgebra in n 1 with a zero coproduct.

Conclusions and outlook
This work has devised a generalisation of the theory of Darboux polynomials to determine and to classify up to Lie algebra automorphisms the coboundary real Lie bialgebras over indecomposable real four-dimensional Lie algebras in a geometric manner. As a byproduct, a technique for the matrix representation of Lie algebras with non-trivial kernel has been developed. Such matrix representations are frequently useful in calculations. The procedures devised in this work are good enough to make affordable the classification, up to Lie algebra automorphisms, of coboundary Lie bialgebras on real and complex, of at least dimension five, indecomposable Lie algebras via theŠnobl and Winternitz classification [50]. The case of two-and three-dimensional Lie bialgebras can also be obtained. Moreover, a brief analysis of the classification of coboundary coproducts on real four-dimensional decomposable Lie algebras through our methods shows that their classification relies partially on the classification of coboundary coproducts on three-and two-dimensional Lie algebras, while the complexity of the procedure is significantly easier than in the present work [25,35]. We expect to tackle this task in the future and to compare our results with previous works on the topic [11,35,30]. Our techniques are also expected to be applied to other higher-dimensional Lie algebras of particular types, e.g. five-dimensional nilpotent or semi-simple Lie algebras. In all previously mentioned cases, we aim to inspect new properties of the Darboux families used to study these problems.
The determination of coboundary Lie bialgebras is relevant to the study of Poisson Lie groups. In particular, it represents an initial step in the study of Poisson Lie groups of dimension four, which would give rise to an extension of the paper [3].
Our work has focused on the classification of coboundary Lie bialgebras on indecomposable Lie algebras. Probably, one laborious task in the application of our method is the determination of a representative of each connected part of the Lie group of Lie algebra automorphisms of a Lie algebra. In the case of semi-simple Lie algebras, this is much easier, as algebraic techniques based on Dynkin diagrams and other results can be applied [13,40]. It is left for further works to study this problem and to search for coboundary Lie bialgebras in the Lie algebra so(3, 2), which has applications in the analysis of quantum gravity in 3+1 dimensions (see [9] and references therein).
Repr. element   Table 2: Orbits of the action of Aut(g) on Y g , their dimensions, elements, and representatives, for real fourdimensional indecomposable Lie algebras g. Each Lie algebra is divided into several subsets enumerated by roman numbers, which classify the orbits of Aut(g) in Y g . The orbits of Aut c (g) are given by the (topologically) connected components of each orbit of Aut(g). The trivial orbits given by 0 ∈ Λ 2 g are not considered. Symbol A ± stands for two orbits, one with + and other with −. In these cases, R ± stands for R + for the first orbit and R − for the second.
If an orbit of Aut(g) in Λ 2 g belongs to Y g only for a certain set of parameters, each family of possible values of the parameters is indicated in a subindex, first, or as a superindex, if a second family of parameters is available. A star ( * ) is used to denote r-matrices that are not solutions to the CYBE.  Table 3: Schouten brackets between basis elements of g and Λ 2 g for real four-dimensional indecomposable Lie algebras.    Table 5: Schouten brackets between basis elements of g and Λ 3 g for real four-dimensional indecomposable Lie algebras.