Remark 7

Canonical indexing. If B and $B'$ are two bases of R, there exists a unique element $w \in W$ such that $w(B) = B'$. We have

for $\alpha, \beta \in B$. Consequently, the Cartan and Coxeter matrices associated to B can be obtained from those associated to $B'$ by composition with the bijection

from B to $B'$.

The Cartan and Coxeter matrices can actually be defined canonically in the following way. Let X be the set of pairs $(B, \alpha)$, where B is a basis of R and $\alpha \in B$. The group W acts in an obvious way on X and each orbit of W on X meets each of the sets $\{B\} \times B$ in exactly one point. If I is the set of these orbits, each basis B admits a canonical indexing $(\alpha_i)_{i \in I}$. Moreover, there exists a unique matrix $N = (n_{ij})$ (resp. $M = (m_{ij})$), of type $I \times I$, such that for any basis B, the Cartan (resp. Coxeter) matrix associated to B can be obtained from N (resp. M) by composing with the canonical indexing of B; it is called the canonical Cartan matrix (resp. Coxeter matrix) of R.