Malcev-like binary Lie algebras of dimension 5

Abstract

We study anti-commutative algebras, which are extensions of a one-dimensional algebra by a four-dimensional nilpotent Lie algebra and at the same time extensions of the two-dimensional non-Abelian Lie algebra by an abelian algebra. These algebras, just like the five-dimensional solvable Malcev algebras, have a flag of subalgebras and can therefore be considered their closest relatives. We characterize binary Lie algebras in this class, called Malcev-like algebras, playing interesting role in non-associative Lie theory. We find normal forms of their multiplications, and determine their isomorphism classes.