= Solution
Restrict the Coxeter matrix of $(W,S)$ to $T\times T$ and let $W'$ be the Coxeter group defined by that matrix. Sending its generators to the corresponding elements of $T$ gives a surjection $W'\to W_T$. The <Geometric representation of a Coxeter group> for $W'$ 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
$$
\boxed{(W_T,T)\text{ is a Coxeter system}.}
$$
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