On Equality of Certain Derivations of Lie Algebras

Abstract

Abstract Let L be a Lie algebra. A derivation α of L is a commuting derivation (central derivation), if α (x) ∈ CL (x) (α (x) ∈ Z (L)) for each x ∈ L. We denote the set of all commuting derivations (central derivations) by 𝒟 (L) (Derz (L)). In this paper, first we show 𝒟 (L) is subalgebra from derivation algebra L, also we investigate the conditions on the Lie algebra L where commuting derivation is trivial and finally we introduce the family of nilpotent Lie algebras in which Derz (L) = 𝒟 (L).