Malcev algebras with $J_{2}$-potent radical

Abstract:

Let $A$ be a Malcev algebra, $B$ be an ideal of $A$ and $J_2^1(B) = J(B,A,A)$ where $J(B,A,A)$ is the linear subspace of $A$ spanned by all elements of the form $J(x,y,z) = (xy)z + (yz)x + (zx)y,x \in B,y,z \in A$. For $k \geq 1$, define $J_2^{k + 1}(B) = J(J_2^k(B),A,A)$. Then $B$ is called ${J_2}$-potent if there exists an integer $N \geq 1$ such that $J_2^N(B) = 0$. Now let $A$ be a Malcev algebra over a field of characteristic 0 such that the radical $R$ of $A$ is ${J_2}$-potent. Then $R$ is complemented by a semisimple subalgebra and all such complements are strictly conjugate in $A$. The proofs follow those in the Lie algebra case.