Corollary 2

Let $\varphi$ be a map from $R$ to a abelian group $\Gamma$ having the following properties:

1) $\varphi(-\alpha) = -\varphi(\alpha)$ for $\alpha \in R$; 2) if $\alpha \in R, \beta \in R$ are such that $\alpha + \beta \in R$, then $\varphi(\alpha + \beta) = \varphi(\alpha) + \varphi(\beta)$.

Let $Q$ be the subgroup of $V$ generated by $R$. Then $\varphi$ extends to a homomorphism from $Q$ to $\Gamma$.

Let $B$ be a basis of $R$. Let $\psi$ be the unique homomorphism from $Q$ to $\Gamma$ that coincides with $\varphi$ on $B$. It suffices to show that $\psi(\alpha) = \varphi(\alpha)$ when $\alpha$ is a positive root relative to $B$. We have $\alpha = \beta_1 + \cdots + \beta_m$ with $\beta_i \in B$ for all $i$, and $\beta_1 + \cdots + \beta_h \in R$ for all $h$ (Prop. 19). We show that $\psi(\alpha) = \varphi(\alpha)$ by induction on $m$. This is clear if $m = 1$. The induction hypothesis gives

and we have $\psi(\beta_m) = \varphi(\beta_m)$, hence $\psi(\alpha) = \varphi(\alpha)$, which proves the corollary.

For any root $\alpha = \sum_{\beta \in B(C)} n_\beta \beta$ in $R$, denote by $Y(\alpha)$ the set of $\beta \in B(C)$ such that $n_\beta \neq 0$. Moreover, observe that $B(C)$ can be identified with the set of vertices of the graph of the Coxeter system formed by $W(R)$ and the $s_{\alpha_i}$ (cf. Chap. IV, § 1, no. 9 and Chap. V, § 3, no. 2).