Remark 2

The root system $R^{\vee}$ in $\mathfrak{h}$ defines a non-degenerate symmetric bilinear form $\beta$ on $\mathfrak{h}$ (Chap. VI, §1, no. 1, Prop. 3), namely the form $(a, b) \rightarrow$ $\sum_{\alpha\in R}\langle \alpha , a\rangle \langle \alpha , b\rangle$. By the Cor. to Th. 1, this form is just the restriction of the

Killing form to $\mathfrak{h}$. The extension of $\beta |\mathfrak{h}_{\mathbf{Q}}\times \mathfrak{h}_{\mathbf{Q}}$ to $\mathfrak{h}_{\mathbf{Q}}\otimes_{\mathbf{Q}}\mathbf{R}$ is positive non-degenerate (Chap. VI, §1, no. 1, Prop. 3). On the other hand, we see that the inverse form on $\mathfrak{h}^*$ of the restriction to $\mathfrak{h}$ of the Killing form on $\mathfrak{g}$ is the canonical bilinear form $\Phi_R$ of R (Chap. VI, §1, no. 12).

Let $(\mathfrak{g}_1,\mathfrak{h}_1), (\mathfrak{g}_2,\mathfrak{h}_2)$ be split semi-simple Lie algebras, $\varphi$ an isomorphism from $\mathfrak{g}_1$ to $\mathfrak{g}_2$ such that $\varphi (\mathfrak{h}_1) =\mathfrak{h}_2$. By transport of structure, the transpose of the map $\varphi |\mathfrak{h}_1$ takes $R(\mathfrak{g}_2,\mathfrak{h}_2)$ to $R(\mathfrak{g}_1,\mathfrak{h}_1)$.