Remark

Let $W_1, \ldots, W_m$ be essential finite groups acting in the spaces $V_1, \ldots, V_m$ and generated by reflections. Let $C_j$ be a chamber relative to $W_j$. Let $W$ be the group $W_1 \times \cdots \times W_m$ acting in the space $V_1 \times \cdots \times V_m$. Then $C_1 \times \cdots \times C_m$ is a chamber relative to $W$. The Coxeter transformations of $W$ defined by $C$ are the products $c_1 c_2 \ldots c_m$, where $c_j$ is a Coxeter transformation of $W_j$ defined by $C_j$.