Naive Lie Theory
《Naive Lie Theory》 · 印刷第 132 页 Chapter 8 的群、根数据、根子群及 Weyl 群约定（含 Proposition 8.1.1 完整陈述）；第 137 页 Section 8.2 的 Borel 子群、幺幂根与正根约定；第 142—144 页的根集合、最长元、子群及代表元约定（含 Lemma 8.3.5 完整陈述）；第 145 页 Corollary 8.3.9 完整陈述
$G$ denotes a connected, reductive, linear algebraic group and $T$ is a maximal torus of $G$. […] Let $(X,R,X^\vee,R^\vee)$ be the root datum of $(G,T)$ (see 7.4.3). 8.1.1. Proposition. (i) For $\alpha\in R$ there exists an isomorphism $u_\alpha$ of $\mathbf G_a$ onto a unique closed subgroup $U_\alpha$ of $G$ such that $tu_\alpha(x)t^{-1}=u_\alpha(\alpha(t)x)$ $(t\in T,x\in k)$. We have $\operatorname{Im}du_\alpha=\mathfrak g_\alpha$, the weight space for the weight $\alpha$ of $T$; (ii) $T$ and the $U_\alpha$ $(\alpha\in R)$ generate $G$. […] Let $W=N_G(T)/T$ be the Weyl group of $(G,T)$ (see 7.1.4). […] Let $B$ be a Borel subgroup of $G$ containing $T$. We denote its unipotent radical by $B_u$ or $U$ and write $R^+$ for the system of positive roots $R^+(B)$ defined by $B$ (see 7.4.6). […] For $w\in W$ put $R(w)=\{\alpha\in R^+\mid w.\alpha\in-R^+\}$. […] By 8.2.4 (ii) there is a unique element $w_0\in W$ with $w_0.R^+=-R^+$. […] 8.3.5. Lemma. Let $w\in W$. (i) The groups $U_\alpha$ with $\alpha\in R(w)$ generate a closed, connected subgroup $U_w$ of $U$ that is normalized by $T$. We have $U_w=\prod_{\alpha\in R(w)}U_\alpha$ (the product being taken in any order); (ii) The product morphism $U_w\times U_{w_0w}\to U$ is an isomorphism of varieties. […] Let $(\dot w)_{w\in W}$ be a set of representatives in $N_G(T)$ of the elements of $W$. […] 8.3.9. Corollary. [Bruhat decomposition] An element of $G$ can be written uniquely in the form $u\dot wb$ with $w\in W$, $u\in U_{w^{-1}}$, $b\in B$.