Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 46 页 · Chapter I, Section 6「Nilpotent Lie Algebras and Engel's Theorem」，Theorem 1.35
Theorem 1.35 (Engel's Theorem). Let $V\ne0$ be a finite-dimensional vector space over $k$, and let $\mathfrak g$ be a Lie algebra of nilpotent endomorphisms of $V$. Then (a) $\mathfrak g$ is a nilpotent Lie algebra, (b) there exists $v\ne0$ in $V$ with $X(v)=0$ for all $X\in\mathfrak g$, (c) in a suitable basis of $V$, all $X$ are upper triangular with $0$'s on the diagonal.