Why are the conjugated generating reflections the only reflections of a finite reflection group?
Suppose $W$ is a finite reflection group. (i.e $W$ is finite and is generated by a set of orthogonal reflections of euclidean space). I want to show that if a set of reflections $T$ generates $W$ then $\{wtw^{-1} | w \in W, t \in T\}$ consists of $\textbf{all}$ reflections of $W$.