Engel's theorem

Let $𝔤$ be a finite-dimensional Lie algebra. Then $𝔤$ is a nilpotent Lie algebra iff all elements are $a d$ -nilpotent.¹ #m/thm/lie

Proof

Let $𝔤 [0]$ be a Lie algebra with all elements $a d$ -nilpotent. Since $a d 𝔤 [0] \leq 𝔤 𝔩 (𝔤 [0])$ is a Lie algebra of nilpotent endomorphisms, there exists some nonzero $𝑥 \in 𝔤 [0]$ such that $[𝔤 [0], 𝑥] = 0$ , i.e. the centre $𝔷 (𝔤 [0]) \neq 0$ .

Thus $𝔤 [1] = 𝔤 [0] / 𝔷 (𝔤 [0])$ has all elements $a d$ -nilpotent and is of smaller dimension. We can repeat the process, and eventually it must bottom out with $𝔷 (𝔤 [𝑘]) = 𝔤 [𝑘]$ ; thus by ^P2 $𝔤 [𝑘 - 1]$ is nilpotent, and so on all the way back to $𝔤 [0]$ .

For the converse, assume $𝔤$ is nilpotent, say $𝔤 𝑛 = 0$ . Then $(a d 𝑥) 𝑛 = 0$ , as required.

#state/tidy | #lang/en | #SemBr

1972. Introduction to Lie Algebras and Representation Theory, §3.3, pp. 12–13 ↩

Engel's theorem

Footnotes