Finite roots and finite Coxeter groups at finite rank

If $W_L$ is finite, the expression ${\mathrm{\Phi }}_L=\{w{\alpha }_i:w\in W_L,i\in L\}$ exhibits ${\mathrm{\Phi }}_L$ as the image of the finite set $W_L\times L$. If $L$ is empty, ${\mathrm{\Phi }}_L$ is empty and $W_L$ is the identity group, so both are finite.

For $i\in L$ and $v\in V_L$, the ambient reflection formula gives $s_{i}v=v-2B(v,{\alpha }_i){\alpha }_i\in V_L$. Since $s_i^2=1$, each such generator preserves $V_L$ bijectively, and hence $W_L$ acts on $V_L$. The restricted form $B{\mathrm{\mid }}_{V_L\times V_L}$ has exactly the Coxeter-form entries for $m_L$. Under the basis identification sending the canonical vector indexed by $i$ to ${\alpha }_i$, these generator actions are therefore the canonical simple reflections for $m_L$. The isomorphism in [F1] consequently identifies the restricted action with that canonical representation on every group element. Since $L$ is finite, [F2] makes the restricted action faithful.

Conversely, suppose ${\mathrm{\Phi }}_L$ is finite. For $w,u\in W_L$ and $i\in L$, one has $w(u{\alpha }_i)=(wu){\alpha }_i\in {\mathrm{\Phi }}_L$; applying the same observation to $w^{-1}$ shows that $w$ permutes ${\mathrm{\Phi }}_L$. This gives a homomorphism from $W_L$ to the finite permutation group of ${\mathrm{\Phi }}_L$. Every ${\alpha }_i$ for $i\in L$ belongs to ${\mathrm{\Phi }}_L$, by taking $u=1$. An element of the kernel thus fixes the basis of $V_L$ pointwise and, by linearity, acts identically on $V_L$. The faithfulness established in step 1.2 forces that element to be the identity. The permutation homomorphism is injective, so $W_L$ is finite. Together with step 1.1 this proves the equivalence.