On semisimple Mal′cev algebras

Abstract

Let M M be a finite dimensional semisimple Malcev algebra over a perfect field of characteristic ≠ 2 , 3 e 2,3 . Let N ( M ) N(M) be its J J -nucleus and J ( M , M , M ) J(M,M,M) the subspace spanned by its jacobians. Then it is shown that M = N ( M ) ⊕ J ( M , M , M ) , N ( M ) M = N(M) \oplus J(M,M,M),N(M) is a semisimple Lie algebra and J ( M , M , M ) J(M,M,M) is a direct sum of simple non-Lie Malcev algebras.