Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 145 页 Chapter VII 的对象约定，第 146—147 页 Section 25.2「Existence of a Chevalley basis」的 Proposition 完整陈述、定义句与 Theorem (Chevalley) 完整陈述
$L$ is a semisimple Lie algebra over the algebraically closed field $F$ of characteristic $0$, $H$ a CSA, $\Phi$ the root system. […] Proposition. It is possible to choose root vectors $x_\alpha\in L_\alpha$ $(\alpha\in\Phi)$ satisfying: (a) $[x_\alpha x_{-\alpha}]=h_\alpha$. (b) If $\alpha,\beta,\alpha+\beta\in\Phi$, $[x_\alpha x_\beta]=c_{\alpha\beta}x_{\alpha+\beta}$, then $c_{\alpha\beta}=-c_{-\alpha,-\beta}$. For any such choice of root vectors, the scalars $c_{\alpha\beta}$ $(\alpha,\beta,\alpha+\beta\in\Phi)$ automatically satisfy: (c) $c_{\alpha\beta}^2=q(r+1)\frac{(\alpha+\beta,\alpha+\beta)}{(\beta,\beta)}$, where $\beta-r\alpha,\ldots,\beta+q\alpha$ is the $\alpha$-string through $\beta$. […] We are now in a position to construct a Chevalley basis of $L$. This is by definition any basis $\{x_\alpha,\alpha\in\Phi;h_i,1\leq i\leq\ell\}$ for which the $x_\alpha$ satisfy (a) (b) of the preceding proposition, while $h_i=h_{\alpha_i}$ for some base $\Delta=\{\alpha_1,\ldots,\alpha_\ell\}$ of $\Phi$. Theorem (Chevalley). Let $\{x_\alpha,\alpha\in\Phi;h_i,1\leq i\leq\ell\}$ be a Chevalley basis of $L$. Then the resulting structure constants lie in $\mathbb Z$. More precisely: (a) $[h_i h_j]=0$, $1\leq i,j\leq\ell$. (b) $[h_i x_\alpha]=\langle\alpha,\alpha_i\rangle x_\alpha$, $1\leq i\leq\ell$, $\alpha\in\Phi$. (c) $[x_\alpha x_{-\alpha}]=h_\alpha$ is a $\mathbb Z$-linear combination of $h_1,\ldots,h_\ell$. (d) If $\alpha,\beta$ are independent roots, $\beta-r\alpha,\ldots,\beta+q\alpha$ the $\alpha$-string through $\beta$, then $[x_\alpha x_\beta]=0$ if $q=0$, while $[x_\alpha x_\beta]=\pm(r+1)x_{\alpha+\beta}$ if $\alpha+\beta\in\Phi$.