Some Remarks on a Classification of Nilpotent Compatible Leibniz Algebras

In this note, we describe compatible Leibniz algebras and present several of their properties. Our aim is to present a comprehensive classification of non-Lie nilpotent compatible Leibniz algebras in low dimensions. By using the full classification of non-Lie nilpotent Leibniz algebras over the complex field of dimensions two, three, and four, in this note, our aim is to provide the classification of non-Lie nilpotent compatible Leibniz algebras over the complex field of dimensions two, three, and four. Consequently, we show that there is no isomorphism class in dimension two, there are 5 isomorphism classes in dimension three, and there are 134 isomorphism classes in dimension four.