Let $W = \langle x_1, x_2, x_3 \;|\; (x_ix_j)^{m_{ij}} \rangle$ be an irreducible Coxeter group, i.e., the graph with vertices $v_1, v_2, v_3$ and edges between all pairs $(v_i, v_j)$ with $m_{ij} \ne 2$, is connected.
Does there always exist a faithful representation $\rho: W \rightarrow \text{SO}(3, \mathbb{R})$? If not, can you find a criterion on $W$ (except for that the reflection representation be positive definite) for which there does exist such a representation?