A Note on Derivations of Lie Algebras

Abstract

In this note, we will prove that a finite dimensional Lie algebra $L$ of characteristic zero, admitting an abelian algebra of derivations $D\le Der(L)$ with the property $$L^n\subseteq \sum _{d\in D}d(L)$$ for some $n\ge 1$, is necessarily solvable. As a result, if $L$ has a derivation $d:L\to L$, such that $L^n\subseteq d(L)$, for some $n\ge 1$, then $L$ is solvable.

4 pages