On special subalgebras of derivations of Lie algebras

Abstract

Let L be a Lie algebra, and Der (L) and IDer (L) be the set of all derivations and inner derivations of L, respectively. Also, let Der c (L) denote the set of all derivations α ∈ Der (L) for which α(x) ∈ Imad x for all x ∈ L. We give necessary and sufficient conditions under which Der c (L) = Der z (L), where Der z (L) is the set of all derivations of L whose images lie in the center of L. Moreover, it is shown that any two isoclinic Lie algebras L 1 and L 2 satisfy Der c (L 1 ) ≅ Der c (L 2 ).