The Structure of D-Derivations and Their Decomposition in Lie Algebras

A D-derivation of a Lie algebra L is a linear map φ for which there exists a derivation D such that φ([x,y])=[φ(x),y]+[x,D(y)] for all x,y∈L. This paper presents explicit structural results concerning D-derivations in Lie algebras over arbitrary fields. It is established that the set of D-derivations forms a Lie algebra, which decomposes as the sum of derivations and centroids, intersecting precisely at the space of central derivations. For centerless Lie algebras, the inclusion chain for D-derivations within existing derivation classes is completed, resulting in a refined hierarchy. It is proven that for both perfect and centerless Lie algebras, D-derivations decompose as a direct sum of derivations and centroids. In particular, for semisimple Lie algebras, it is shown that DerD(L)=ad(L)⊕C(L), and for simple Lie algebras over an algebraically closed field of characteristic zero, DerD(L)=ad(L)⊕FidL. Furthermore, for any centerless Lie algebra, the Lie algebra of D-derivations is shown to be isomorphic to the semidirect product of the derivation and centroid algebras, with explicit descriptions provided for semisimple and solvable cases. Examples involving so(3), so(1,3), aff(1), and h3 confirm these decompositions and offer matrix realizations of their D-derivations, thereby supporting and illustrating the main theorems.