Given a Coxeter group $\varGamma =\langle \rho_0, \rho_1, \ldots, \rho_{n - 1}\rangle$ which at least satisfy the relation $(\rho_i\rho_j)^{p_{ij}}=1, \ \ 0\leq i, j \leq n - 1,$ where $p_{ii}=1$ and $2 \leq p_{ij}\leq \infty$ for all $i\neq j.$ On an n - dimensional real vector space V, with basis $a_0, a_1, \ldots, a_{n - 1},$ we define a symmetric bilinear form $x\cdot y$ by setting $a_i \cdot a_j:=-2 \text{cos} \frac{\pi}{p_{ij}}, \ \ 0\leq i, j, \leq n - 1.$
How do I show bilinearity? Thanks