 Solution

The Weyl reflection in $\alpha$ satisfiesNow take $\beta \in \Pi \setminus \{\alpha \}$ and expand it in the basis $\Delta$. At least one coefficient belonging to a simple root other than $\alpha$ is positive. Sincethe reflection changes only the coefficient of $\alpha$. Every root has simple-root coefficients of one sign, so the unchanged positive coefficient prevents $s_{\alpha }(\beta )$ from being negative. Hence $s_{\alpha }(\beta )\in \Pi$. Because $s_{\alpha }$ is an involution, it permutes $\Pi \setminus \{\alpha \}$, while it exchanges $\alpha$ and $-\alpha$. Therefore

Solved by gpt-5.6-sol high.