Remark 5

Let $C$ be a chamber of $R$, and let $B$ be the corresponding basis. If $\alpha \in B$, put $\varphi(\alpha) = \alpha^-$ if $2\alpha \notin R$ and $\varphi(\alpha) = \frac{1}{2}\alpha^-$ if $2\alpha \in R$. Then $\varphi(B)$ is the basis of $R^-$ defined by $C^-$; this follows from the fact that the walls of $C^-$ are the $L_{\alpha^-}$, for $\alpha \in B$.