Standard Lie identity in solvable varieties of associative type (Q1358576)

Throughout the article, only Lie algebras over a field of characteristic 0 are considered. It is shown by the author in [Vestn. Mosk. Univ., Ser. I 1993, No. 5, 63-66 (1993; Zbl 0840.17008)] that, for solvable of class \(\leq3\) varieties of Lie algebras, the standard identity of degree 6 does not imply a system of Capelli identities of any order. In the article [ibid. 1992, No. 3, 55-57 (1992; Zbl 0779.17003)] the author has introduced the notion of varieties of Lie algebras of associative type. For instance, any special variety has associative type. Theorem 1 of the article under consideration states that a solvable variety of Lie algebras of associative type satisfies a standard identity if and only if it satisfies a system of Capelli identities of some order. It is shown in the article by \textit{M. V. Zaitsev} [Vestn. Mosk. Univ., Ser. I 1993, No. 1, 56-59 (1993; Zbl 0817.17006)] that an analogous result for non-solvable (even special) varieties fails. Theorem 2 states that a subvariety of associative type in a variety generated by an infinite dimensional simple algebra of Cartan type satisfies a standard identity if and only if it satisfies a system of Capelli identities of some order. Theorem 2 is given without detailed proof.