Remark 2.4.7.4. Let $(W,S)$ be a Coxeter system. Then the length function $\ell : W \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$ has the following properties:
An element $w \in W$ satisfies $\ell (w) = 0$ if and only if $w = 1$ is the identity element of $W$.
An element $w \in W$ satisfies $\ell (w) = 1$ if and only if $w$ belongs to $S$.
For every pair of elements $w,w' \in W$, we have $\ell ( w w' ) \leq \ell (w) + \ell (w')$. Moreover, we also have $\ell ( w w' ) \equiv \ell (w) + \ell (w') \pmod{2}$.