Definition 1

Let $V$ be a vector space over $k$, and $R$ a subset of $V$. Then $R$ is said to be a root system in $V$ if the following conditions are satisfied:

(RS_I) $R$ is finite, does not contain 0, and generates $V$. (RS_{II}) For all $\alpha \in R$, there exists an element $\alpha^\vee$ of the dual $V^*$ of $V$ such that $\langle \alpha, \alpha^\vee \rangle = 2$ and that the reflection $s_{\alpha, \alpha^\vee}$ (cf. Chap. V, § 2) leaves $R$ stable.

(RS_{\text{III}}) For all $\alpha \in \mathbf{R}$, $\alpha^{\sim}(R) \subseteq \mathbf{Z}$.

By Lemma 1, the reflection $s_{\alpha,\alpha^{\sim}}$ (and hence also the linear form $\alpha^{\sim}$) is determined uniquely by $\alpha$, so (RS_{\text{III}}) makes sense. We put $s_{\alpha,\alpha^{\sim}} = s_{\alpha}$. Then $s_{\alpha}(x) = x - \langle \alpha^{\sim}, x \rangle \alpha$ for all $x \in V$.

The elements of R are called the roots (of the system considered). The dimension of V is called the rank of the system.

The automorphisms of V that leave R stable are called the automorphisms of R. They form a finite group denoted by A(R). The subgroup of A(R) generated by the $s_{\alpha}$ is called the Weyl group of R and is denoted by W(R), or simply by W.