A Note on Derivations of Lie Algebras

In this note, we will prove that a finite dimensional Lie algebra $L$ over an algebraically closed field of characteristic zero, admitting an abelian algebra of derivations $D\leq Der(L)$ with the property $$ L^n\subseteq \sum_{d\in D}d(L) $$ for some $n>1$, is necessarily solvable. As a result, we show that, if $L$ has a derivation $d:L\to L$, such that $L^n\subseteq d(L)$, for some $n>1$, then $L$ is solvable. doi:10.1017/S000497271100251