5-Dimensional Malcev-Like Algebras

Abstract The five-dimensional anti-commutative algebras having an analogous family of flags of subalgebras as the solvable Malcev algebras form the class of Malcev-like algebras. Recently a classification of the binary Lie algebras in this class is achieved. Our investigation extends this result. We determine Malcev-like algebras over a field $${\mathbb {K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> of characteristic zero. These algebras are extensions of $${\mathbb {K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> by a 4-dimensional nilpotent Lie algebra and simultaneously semidirect sums of the two-dimensional non-abelian Lie algebra and an abelian algebra. We find normal forms of their multiplications and describe their isomorphism classes.