Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 187–188 页 · Chapter II，Section 9「Serre Relations」，Proposition 2.95 与 Remark
Proposition 2.95. The set $X = \{h_i, e_i, f_i\}_{i=1}^l$ satisfies the following properties within $\mathfrak g$: (a) $[h_i, h_j] = 0$, (b) $[e_i, f_j] = \delta_{ij} h_i$, (c) $[h_i, e_j] = A_{ji} e_j$, (d) $[h_i, f_j] = -A_{ji} f_j$, (e) $(\operatorname{ad} e_i)^{-A_{ji}+1} e_j = 0$ when $i \ne j$, (f) $(\operatorname{ad} f_i)^{-A_{ji}+1} f_j = 0$ when $i \ne j$. Remark. Relations (a) through (f) are called the Serre relations for $\mathfrak g$.