Example 2

If $D = (M,M_0,R)$ is a root diagram, let $M^*_0$ be the orthogonal complement of V(R) in $M^*$, and let $R^{\vee}$ be the set of inverse roots of D. Then $D^{\vee}= (M^*,M^*_0,R^{\vee})$ is a root diagram, called the inverse of D. For all $\alpha \in R$, the symmetry $s_{\alpha^{\vee}}$ of $M^*$ is the contragredient automorphism of the symmetry $s_{\alpha}$ of M; the map $w \rightarrow^tw^{-1}$ is an isomorphism from W(D) to $W(D^{\vee})$. Moreover, $V(R^{\vee})$ can be naturally identified with the dual of the $\mathbf{Q}$-vector space $V(R), R^{\vee}$ then being identified with the inverse root system of R.

If the dual of $M^*$ is identified with M, the inverse diagram of $D^{\vee}$ is identified with D.