Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 印刷第 626 页 Chapter X Section 3「Vinberg's Theorem」，开头的整数分次定义与 Lemma 10.15 完整陈述
A complex semisimple Lie algebra $\mathfrak g$ is said to be graded if vector subspaces $\mathfrak g^k$ are specified such that $\mathfrak g=\bigoplus_{k=-\infty}^{\infty}\mathfrak g^k$ and $[\mathfrak g^j,\mathfrak g^k]\subseteq\mathfrak g^{j+k}$ for all integers $j$ and $k$. In other words, $\mathfrak g$ is to be graded as a vector space in the sense of (A.35), and the grading is to be consistent with the bracket structure. Since $\mathfrak g$ is by assumption finite dimensional, $\mathfrak g^k$ has to be $0$ for all but finitely many $k$. […] Lemma 10.15. If $\mathfrak g=\bigoplus_k\mathfrak g^k$ is a graded complex semisimple Lie algebra, then there exists $H$ in $\mathfrak g^0$ such that $\mathfrak g^k=\{X\in\mathfrak g\mid[H,X]=kX\}$ for all $k$.