### A Note on Derivations of Lie Algebras

#### Abstract

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/S0004972711002516

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/S0004972711002516

#### Keywords

Lie algebras, Derivations, Solvable Lie algebras, Compact Lie groups

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Bulletin of the Aust. Math. Soc., copyright Australian Mathematical Publishing Association Inc.