Example 3

Let $(\mathfrak{g},\mathfrak{h})$ be a split reductive $\mathbf{Q}$-Lie algebra, and $M\subset \mathfrak{h}$ a permissible lattice (Chap. VIII, §12, no. 6, Def. 1). Let $M_0$ be the subgroup of M orthogonal to the roots of $(\mathfrak{g},\mathfrak{h})$ and $R^{\vee}$ the set of the $H_{\alpha},\alpha \in R(\mathfrak{g},\mathfrak{h})$. Then $(M,M_0,R^{\vee})$ is a root diagram, and $(M^*,M^*_0,R(\mathfrak{g},\mathfrak{h}))$ is the inverse diagram.