Definition 2

A root diagram (or simply a diagram, if there is no risk of confusion) is a triple $D = (M,M_0,R)$ where:

(RD$_0$) $M$ is a free $\mathbf{Z}$-module of finite type and the submodule $M_0$ is a direct factor of M;

(RD$_I$) $R$ is a finite subset of $M; R\cup M_0$ generates the $\mathbf{Q}$-vector space $\mathbf{Q}\otimes M$;

(RD$_{II}$) for all $\alpha \in R$, there exists an element $\alpha^{\vee}$ of $M^*=$ Hom$_{\mathbf{Z}}(M,\mathbf{Z})$ such that $\alpha^{\vee}(M_0) = 0,\alpha^{\vee}(\alpha ) = 2$ and the endomorphism $x \rightarrow x-\alpha^{\vee}(x)\alpha$ of M leaves R stable.

By Chap. VI, §1, no. 1, for all $\alpha \in R$ the element $\alpha^{\vee}$ of $M^*$ is uniquely determined by $\alpha$; we denote by $s_{\alpha}$ the endomorphism $x \rightarrow x-\alpha^{\vee}(x)\alpha$ of M. Moreover (loc. cit.), the $\mathbf{Q}$-vector space $\mathbf{Q}\otimes M$ is the direct sum of $\mathbf{Q}\otimes M_0$ and the vector subspace V(R) generated by R, and R is a root system in V(R) (loc. cit., Def. 1).

The elements of R are called the roots of the root diagram D, and the elements $\alpha^{\vee}$ of $M^*$ the inverse roots. The group generated by the automorphisms $s_{\alpha}$ of M is called the Weyl group of D and is denoted by W(D); the elements of W(D) induce the identity on $M_0$, and induce on V(R) the transformations of the Weyl group of the root system R.