Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \mapsto -w \cdot \alpha$ if $w \cdot \alpha \in \Phi^-$, where $\Phi^-$ is the set of negative roots.
The group $W$ acts on $(\Phi^+ \times \Phi^+) \backslash \{(\alpha, \alpha): \alpha \in \Phi^+\}$ by $w\cdot (\alpha, \beta) = (w \cdot \alpha, w \cdot \beta)$. Is this action transitive? That is, for $(\alpha, \beta)$, $(\alpha', \beta') \in (\Phi^+ \times \Phi^+) \backslash \{(\alpha, \alpha): \alpha \in \Phi^+\}$, is there $w \in W$ such that $w \cdot(\alpha, \beta) = (\alpha', \beta')$?
In type $A_2$, this action is transitive: let $s_1, s_2$ be the simple reflections and $\alpha_1, \alpha_2$ be simple roots. Then \begin{align} & s_1 \cdot (\alpha_1, \alpha_2) = (\alpha_1, \alpha_1+\alpha_2), \ s_2 (\alpha_1, \alpha_1+\alpha_2)=(\alpha_1+\alpha_2, \alpha_1), \\ & s_2(\alpha_1, \alpha_2) = (\alpha_1+\alpha_2, \alpha_2), s_1(\alpha_1+\alpha_2, \alpha_2) = (\alpha_2, \alpha_1+\alpha_2), \\ & s_2(\alpha_2, \alpha_1+\alpha_2)=(\alpha_2, \alpha_1). \end{align} Is it true in general? Are there some references about this? Thank you very much.
