Let $W$ be a finite reflection group with irreducible root system $\Phi$. Let $\Delta$ be the set consisting of reflections $s_{\alpha}$ corresponding to pairs $\{\alpha, -\alpha \}$ of roots in $\Phi$. $W$ acts on $\Delta$ by conjugation.
I found this action is transitive under the case of $A_2$, is the action transitive under the case of $A_n$? and transitive under other finite cases? Thank you.