I am working through the book Reflection Groups and Coxeter Groups by Humphreys. I got stuck while trying Exercise 1.12.3:
If $w \in W$ is an involution, prove that $w$ can be written as a product of commuting reflections.
The hint given says to use induction on the dimension of $V$ (the vector space on which $W$ acts). Here's what I've been trying: the base case is trivial. Suppose we know that the statement holds for any reflection group acting on a space of dimension $n$. Suppose now that $W$ is a reflection group acting on a space of dimension $n + 1$; therefore, the root system corresponding to $W$ has a simple system $\Delta = \{\alpha_1, \dots, \alpha_{n+1}\}$ consisting of $n + 1$ roots. Set $S := \{s_i \mid I \in [n]\}$ to be the set of simple reflections, where $s_i := s_{\alpha_i}$ (the simple reflection corresponding to $\alpha_i$). An involution $w \in W$ can be written as a reduced word as $w = s_{n_1}\dots s_{n_k}$. Since $w^2 = 1$, any cyclic permutation of this word for $w$ also gives an involution.
I haven't been able to figure out how to use induction here. Could someone point me in the right direction?
