Kronecker factorization theorems for the exceptional Malcev algebra

Abstract

We prove that a Malcev algebra $\mathcal{M}$ containing the $7$-dimensional simple non-Lie Malcev algebra $\mathbb{M}$ such that $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}$, is isomorphic to $\mathbb{M}\otimes_\textup{F} \mathcal{U}$, where $\mathcal{U}$ is a certain commutative associative algebra. Also, we prove that a Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ whose even part $\mathcal{M}_0$ contains $\mathbb{M}$ with $m\mathbb{M}\neq 0$ for any homogeneous element $0\neq m\in \mathcal{M}_0\cup \mathcal{M}_1$, is isomorphic to $\mathbb{M}\otimes_\textup{F}U$, where $U$ is a certain supercommutative associative superalgebra.