Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 663 页 · Appendix B, Section 3「Ado's Theorem」，Theorem B.8
Theorem B.8 (Ado's Theorem). Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb R$, let $\operatorname{rad}\mathfrak g$ be its radical, and let $\mathfrak n$ be its unique largest nilpotent ideal given as in Corollary 1.41. Then there exists a one-one finite-dimensional representation $\varphi$ of $\mathfrak g$ on a complex vector space such that $\varphi(Y)$ is nilpotent for every $Y$ in $\mathfrak n$. If $\mathfrak g$ is complex, then $\varphi$ can be taken to be complex linear.