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The Coxeter Gram matrix is the Gram matrix $G=(-2\cos (\pi /m_{ij}){)}_{i,j\in I}$ of the distinguished basis in the Geometric representation of a Coxeter group.

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

Let $V$ be the real vector space with basis $(e_i{)}_{i\in I}$. The Coxeter Gram matrix defines the symmetric bilinear formIts diagonal entries are $2$. The Geometric representation of a Coxeter group is generated by the reflectionsEach has square one, and on $\mathrm{span}\{e_i,e_j\}$ the product $\sigma (x_i)\sigma (x_j)$ has order $m_{ij}$. The reflections therefore satisfy the Coxeter relations and define a group representation $\sigma :W\to GL(V)$.

SetIn the Geometric representation of a Coxeter group,including $i=j$, since $a_{ii}=2\cos \pi =-2$. As ${\beta }_i=\sigma (c_{i-1})e_i$, telescoping givesSumming from $i=1$ to $j-1$ yieldsor equivalentlySumming the same telescoping identity all the way to $n$ and substituting this first formula giveswhich is the second required identity.