A Note on Derivations of Lie Algebras

In this note, we will prove that a finite dimensional Lie algebra $L$ 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 $n\geq 1$, is necessarily solvable. As a result, if $L$ has a derivation $d:L\to L$, such that $L^n\subseteq d(L)$, for some $n\geq 1$, then $L$ is solvable.

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