Theorem 2

(i) The set $R = R(\mathfrak{g},\mathfrak{h})$ is a reduced root system in $\mathfrak{h}^*$.

(ii) Let $\alpha \in R$. The map $s_{\alpha ,H_{\alpha}}:\lambda \rightarrow \lambda -\lambda (H_{\alpha})\alpha$ from $\mathfrak{h}^*$ to $\mathfrak{h}^*$ is the unique reflection $s$ of $\mathfrak{h}^*$ such that $s(\alpha ) =-\alpha$ and $s(R) = R$. For all $t\in k^*$, $s$ is the transpose of $\theta_{\alpha}(t)|\mathfrak{h}$.

First, R generates $\mathfrak{h}^*$, for if $h\in \mathfrak{h}$ is such that $\alpha (h) = 0$ for all $\alpha \in R$, then ad $h= 0$ and hence $h= 0$ since the centre of $\mathfrak{g}$ is zero. By definition, $0\notin R$. Let $\alpha \in R$. Since $\alpha (H_{\alpha}) = 2,s=s_{\alpha ,H_{\alpha}}$ is a reflection such that $s(\alpha ) =-\alpha$. Then $s(R) = R$ by Lemma 1 (iii), and $\beta (H_{\alpha})\in \mathbf{Z}$ for all $\beta \in R$ (Prop. 2 (i)). This shows that R is a root system in $\mathfrak{h}^*$. For all $h\in \mathfrak{h}$ and $\lambda \in \mathfrak{h}^*$,

so $s$ is the transpose of $\theta_{\alpha}(t)|\mathfrak{h}$. Finally, we show that the root system R is reduced. Let $\alpha \in R$ and $y\in \mathfrak{g}^{2\alpha}$. Since $3\alpha \notin R$ (Chap. VI, §1, no. 3, Prop. $8$)$, [X_{\alpha}, y] = 0$; on the other hand, $[X_{-\alpha}, y]\in \mathfrak{g}^{-\alpha+2\alpha}=\mathfrak{g}^{\alpha}=kX_{\alpha}$, so $[X_{\alpha},[X_{-\alpha}, y]] = 0$; thus

so $y= 0$ and $\mathfrak{g}^{2\alpha}= 0$. In other words, $2\alpha$ is not a root. Q.E.D.

Identify $\mathfrak{h}$ canonically with $\mathfrak{h}^{**}$. With the notations of Chap. VI, §1, no. 1, we then have, by Th. 2 (ii),

$H_{\alpha}=\alpha^{\vee}$ for all $\alpha \in R$. (2)

The $H_{\alpha}$ thus form the root system $R^{\vee}$ in $\mathfrak{h}$ inverse to R.

We shall call $R(\mathfrak{g},\mathfrak{h})$ the root system of $(\mathfrak{g},\mathfrak{h})$. The reflections $s_{\alpha ,H_{\alpha}}$ will be denoted simply by $s_{\alpha}$. The Weyl group, group of weights, Coxeter number $. .$. of $R(\mathfrak{g},\mathfrak{h})$ are called the Weyl group, group of weights, Coxeter number $. .$. of $(\mathfrak{g},\mathfrak{h})$. As in Chap. VI, §1, no. 1, we consider the Weyl group as operating not only on $\mathfrak{h}^*$, but also on $\mathfrak{h}$ by transport of structure, so that $s_{\alpha}=\theta_{\alpha}(t)|\mathfrak{h}$. Since the $\theta_{\alpha}(t)$ are elementary automorphisms of $\mathfrak{g}$, we have: