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 $m\mathbb{M}\ne 0$ for any $m\ne 0$ from $\mathcal{M}$, is isomorphic to $\mathbb{M}{\otimes }_{\text{F}}\mathcal{U}$, 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 $m\mathbb{M}\ne 0$ for any homogeneous element $0\ne m\in {\mathcal{M}}_0\cup {\mathcal{M}}_1$, is isomorphic to $\mathbb{M}{\otimes }_{\text{F}}U$, where $U$ is a certain supercommutative associative superalgebra.