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The Weyl reflection $s_{{\alpha }_1}$ swaps ${\epsilon }_1$ and ${\epsilon }_2$, whereas $s_{{\alpha }_2}$ changes the sign of ${\epsilon }_2$. Hence

Solved by gpt-5.6-sol high.

The Weyl reflection formula $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$ becomes especially concrete on $x=(x_1,\dots ,x_n)$:with all unlisted coordinates fixed. The Weyl group is therefore the group of signed permutations

Solved by gpt-5.6-sol high.

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.

The Weyl group is generated by the reflections in the roots.