Definition 2.4.7.1. A Coxeter system is a pair $(W,S)$, where $W$ is a group and $S \subseteq W$ is a subset with the following properties:
Each element of $S$ has order $2$.
For each $s,t \in S$, let $m_{s,t} \in \operatorname{\mathbf{Z}}_{ > 0} \cup \{ \infty \} $ denote the order of the product $st$ in the group $W$. Then the inclusion $S \hookrightarrow W$ exhibits $W$ as the quotient of the free group generated by $S$ by the relations $(st)^{m_{s,t}} = 1$ (indexed by those pairs $(s,t)$ with $m_{s,t} < \infty $).