Proposition 3

Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}_1$ and $\mathfrak{h}_2$ splitting Cartan subalgebras of $\mathfrak{g}$. There exists an isomorphism from $\mathfrak{h}^*_1$ to $\mathfrak{h}^*_2$ that takes $R(\mathfrak{g},\mathfrak{h}_1)$ to $R(\mathfrak{g},\mathfrak{h}_2)$.

(For more precise results, see §3, no. 3, Cor. of Prop. 10, and §5, no. 3, Prop. 5).

Let $k'$ be an algebraic closure of $k,\mathfrak{g}'=\mathfrak{g}\otimes_kk',\mathfrak{h}'_i=\mathfrak{h}_i\otimes_kk'$. Then $R(\mathfrak{g}',\mathfrak{h}'_i)$ is the image of $R(\mathfrak{g},\mathfrak{h}_i)$ under the map $\lambda \rightarrow \lambda \otimes 1$ from $\mathfrak{h}^*_i$ to $\mathfrak{h}^*_i\otimes_kk'={\mathfrak{h}'_i}^*$. By Chap. VII, §3, no. 2, Th. 1, there exists an automorphism of $\mathfrak{g}'$ taking $\mathfrak{h}'_1$ to $\mathfrak{h}'_2$, hence an isomorphism $\varphi$ from ${\mathfrak{h}'_1}^*$ to ${\mathfrak{h}'_2}^*$ that takes $R(\mathfrak{g}',\mathfrak{h}'_1)$ to $R(\mathfrak{g}',\mathfrak{h}'_2)$. Then $\varphi |\mathfrak{h}^*_1$ takes $R(\mathfrak{g},\mathfrak{h}_1)$ to $R(\mathfrak{g},\mathfrak{h}_2)$, and hence $\mathfrak{h}^*_1$ to $\mathfrak{h}^*_2$. Q.E.D.

In view of Prop. 3, the root system of $(\mathfrak{g},\mathfrak{h})$ depends, up to isomorphism, only on $\mathfrak{g}$ and not on $\mathfrak{h}$. In the same way, the Weyl group, group of weights $. .$. of $(\mathfrak{g},\mathfrak{h})$ are simply called, by abuse of language, the Weyl group, group of weights $. .$. of $\mathfrak{g}$ (cf. also §5, no. 3, Remark 2). If the Dynkin graph of $\mathfrak{g}$ is of type $A_l$, or $B_l,. .$. (cf. Chap. VI, §4, no. 2, Th. 3), we say that $\mathfrak{g}$ is of type $A_l$, or $B_l,. .$..

Recall that, if $\alpha$ and $\beta$ are linearly independent roots, the set of $j\in \mathbf{Z}$ such that $\beta +j\alpha \in R$ is an interval $[-q, p]$ of $\mathbf{Z}$ containing 0, with $p-q=$ $-\langle \beta , \alpha^{\vee}\rangle =-\beta (H_{\alpha})$ (Chap. VI, §1, no. 3, Prop. 9).