Malcev-like binary Lie algebras of dimension 5 (Q6636348)

The present paper is a logical continuation of another paper by the same authors, in which the classification of 4-dimensional algebras with the property of \(4\)-dimensional binary Lie algebras was given (see [\textit{Á. Figula} and \textit{P. T. Nagy}, Linear Algebra Appl. 656, 385--408 (2023; Zbl 1514.17005)]). Here, the classification of 5-dimensional algebras with the property of \(5\)-dimensional Malcev algebras is given.\N\NThe variety of Malcev algebras is a generalization of the variety of Lie algebras and, on the other hand, it is a proper subvariety within the variety of binary Lie algebras (i.e., each \(2\)-generated Malcev algebra is a Lie algebra). The classification of \(5\)-dimensional Malcev (non-Lie) algebras was given in a paper by \textit{E. N. Kuz'min} [Algebra Logika 9, 691--700 (1970; Zbl 0244.17018)]. As mentioned by the authors, each 5-dimensional solvable Malcev algebra \((M, \cdot)\) has the following properties:\N\begin{itemize}\N\item[(i)] the derived algebra \(M\cdot M\) is a \(4\)-dimensional nilpotent Lie algebra,\N\item[(ii)] \(M\) is an anti-commutative semidirect sum \( {\mathfrak l}_2 \oplus {\mathfrak i}\) where \({\mathfrak l}_2\) is the \(2\)-dimensional nonabelian Lie algebra and \({\mathfrak i}\) is an abelian ideal.\N\end{itemize}\NThe authors fixed the two indicated properties and described all \(5\)-dimensional binary Lie algebras satisfying these two properties in Section 10.