= Solution
The <Weyl reflection> in $\alpha$ satisfies
$$
s_\alpha(\alpha)=-\alpha.
$$
Now 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. Since
$$
s_\alpha(\beta)=\beta-\langle\beta,\alpha^\vee\rangle\alpha,
$$
the 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
$$
s_\alpha(\Pi)=(\Pi\sqcup\{-\alpha\})\setminus\{\alpha\}.
$$
Solved by gpt-5.6-sol high.
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