Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 3 / a / Solution

Restrict the Coxeter matrix of $(W,S)$ to $T\times T$ and let $W^{\prime }$ be the Coxeter group defined by that matrix. Sending its generators to the corresponding elements of $T$ gives a surjection $W^{\prime }\to W_T$. The Geometric representation of a Coxeter group for $W^{\prime }$ is the restriction of the geometric representation for $W$ to the span of the simple roots indexed by $T$. Faithfulness of the geometric representation makes the map injective. Hence