Exercise 4, § 1

Let $\Gamma$ be a free $\mathbf{Z}$-module of finite rank l, $\Gamma^*$ the dual $\mathbf{Z}$-module, I a finite set of indices, $(x_i, x_i^*)_{i \in I}$ a family of elements of $\Gamma \times \Gamma^*$ such that $\langle x_i, x_i^* \rangle = 2$ for all $i \in I$, $s_i = s_{x_i, x_i^*}$. Let V be the real vector space $\Gamma \otimes_{\mathbf{Z}} \mathbf{R}$, whose dual $V^*$ can be identified with $\Gamma^* \otimes_{\mathbf{Z}} \mathbf{R}$. We also denote by $s_i$ the reflection $s_i \otimes 1$ in V. Let R (resp. $R'$) be the set of $x_i$ (resp. $x_i^*$). Let E (resp. $E'$) be the subspace of V (resp. $V^*$) generated by R (resp. $R'$). Assume that $s_i(R) = R$ for all i. Then, R is a root system in E. $E'$ is canonically identified with the dual of E and $x_i^*$ with $x_i^-$ for all i. If F (resp. $F'$) is the subspace of V (resp. $V^*$) consisting of the points invariant under all the $s_i$ (resp. ${}^t s_i$), then $\Gamma \cap F$ (resp. $\Gamma^* \cap F'$) generates F (resp. $F'$). If we put $\Gamma_1 = (\Gamma \cap E) \oplus (\Gamma \cap F)$ (resp. $\Gamma_1^* = (\Gamma^* \cap E') \oplus (\Gamma^* \cap F')$), then $\Gamma / \Gamma_1$ and $\Gamma^* / \Gamma_1^*$ are isomorphic finite groups.