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On the existence of ad-nilpotent elements. (English) Zbl 0359.17008
This note studies conditions which are sufficient to guarantee that a derivation on a finite dimensional Lie algebra over an algebraically closed field is nilpotent. It is shown that if a derivation is nilpotent on the span of the set of eigenvectors of $$\mathrm{ad}_x$$ for some $$x$$ in the Lie algebra, then it is nilpotent. This result is used to prove the following theorem: Let $$L$$ be a finite dimensional Lie algebra over an algebraically closed field. Let $$X\subseteq L$$ be a non-empty subset such that for each $$x\in L$$ all eigenvectors of $$\mathrm{ad}_x$$ lie in $$X$$. Then $$\mathrm{ad}_y$$ is nilpotent for some $$y\in X$$. A consequence of the theorem when $$X=L\setminus\{0\}$$ is an elementary proof of the fact that every finite dimensional Lie algebra over an algebraically closed field possesses an ad-nilpotent element.
Reviewer: George M. Benkart

##### MSC:
 17B05 Structure theory for Lie algebras and superalgebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 17B50 Modular Lie (super)algebras