Remark 1

If $\mathfrak{h}_{\mathbf{Q}}$ (resp. $\mathfrak{h}^*_{\mathbf{Q}}$) denotes the $\mathbf{Q}$-vector subspace of $\mathfrak{h}$ (resp. $\mathfrak{h}^*$) generated by the $H_{\alpha}$ (resp. the $\alpha$ ), where $\alpha \in R$, then $\mathfrak{h}$ (resp. $\mathfrak{h}^*$) can be identified canonically with $\mathfrak{h}_{\mathbf{Q}}\otimes_{\mathbf{Q}}k$ (resp. with $\mathfrak{h}^*_{\mathbf{Q}}\otimes_{\mathbf{Q}}k$) and $\mathfrak{h}^*_{\mathbf{Q}}$ can be identified with the dual of $\mathfrak{h}_{\mathbf{Q}}$ (Chap. VI, §1, no. 1, Prop. 1). We call $\mathfrak{h}_{\mathbf{Q}}$ and $\mathfrak{h}^*_{\mathbf{Q}}$ the canonical $\mathbf{Q}$-structures on $\mathfrak{h}$ and $\mathfrak{h}^*($Algebra, Chap. II, §8, no. 1, Def. 1). When we mention $\mathbf{Q}$-rationality for a vector subspace of $\mathfrak{h}$, for a linear form on $\mathfrak{h}$, etc., we shall mean these structures, unless we indicate otherwise. When we mention Weyl chambers, or facets, of $R(\mathfrak{g},\mathfrak{h})$, we shall work in $\mathfrak{h}_{\mathbf{Q}}\otimes_{\mathbf{Q}}\mathbf{R}$ or $\mathfrak{h}^*_{\mathbf{Q}}\otimes_{\mathbf{Q}}\mathbf{R}$, that we shall denote by $\mathfrak{h}_{\mathbf{R}}$ and $\mathfrak{h}^*_{\mathbf{R}}$.