Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 15 页 Chapter II 章首域的约定、第 35 页 Section 8「Root space decomposition」章首及 Section 8.1「Maximal toral subalgebras and roots」（省略中间内容）
Virtually all of the remaining theory to be developed in this book will require the assumption that $F$ have characteristic $0$. (Some of the exercises will indicate how counterexamples arise in prime characteristic.) Moreover, in order to have available the eigenvalues of $\operatorname{ad}x$ for arbitrary $x$ (not just for $\operatorname{ad}x$ nilpotent), we shall assume that $F$ is algebraically closed, except where otherwise specified. […] Throughout this section $L$ denotes a (nonzero) semisimple Lie algebra. […] Now fix a maximal toral subalgebra $H$ of $L$, i.e., a toral subalgebra not properly included in any other. […] In other words, $L$ is the direct sum of the subspaces $L_\alpha=\{x\in L\mid[hx]=\alpha(h)x\text{ for all }h\in H\}$, where $\alpha$ ranges over $H^*$. Notice that $L_0$ is simply $C_L(H)$, the centralizer of $H$; it includes $H$, thanks to the lemma. The set of all nonzero $\alpha\in H^*$ for which $L_\alpha\ne0$ is denoted by $\Phi$; the elements of $\Phi$ are called the roots of $L$ relative to $H$ (and are finite in number). With this notation we have a root space decomposition (or Cartan decomposition): (*) $L=C_L(H)\oplus\coprod_{\alpha\in\Phi}L_\alpha$.