On Derivation Algebras of Malcev Algebras and Lie Triple Systems

W. H. Davenport has shown that the derivation algebra 3)(4) of a semisimple Malcev algebra A of characteristic 0 acts completely reducibly on A. The purpose of the present note is to characterize those Malcev algebras which have such derivation algebras as those whose radical is central and to obtain the same result for Lie triple systems. Analogous results are known to hold for standard and alternative algebras. Theorem. Let A be a Malcev algebra or a Lie triple system over a field of characteristic 0. Then the derivation algebra ^(A) of A acts completely reducibly on A if and only if the radical R of A is contained in the center Z of A. All algebras considered here are finite dimensional over a field of characteristic 0 and the unexplained notation is as in [8]. We note that if S is a subset of a vector space A, then <5> denotes the linear span of S. Remark. The referee has noted that for Malcev algebras the condition that the radical is central is characterized in several ways in [5, Lemma 3]. 1. The Malcev algebra case. An algebra A is said to be Malcev if the identities