= Solution
The <Weyl reflection> formula $s_\alpha(x)=x-\langle x,\alpha^\vee\rangle\alpha$ becomes especially concrete on $x=(x_1,\ldots,x_n)$:
$$
\begin{aligned}
s_{\varepsilon_i-\varepsilon_j}&:\ (x_i,x_j)\longmapsto(x_j,x_i),\\
s_{\varepsilon_i+\varepsilon_j}&:\ (x_i,x_j)\longmapsto(-x_j,-x_i),\\
s_{2\varepsilon_i}&:\ x_i\longmapsto-x_i,
\end{aligned}
$$
with all unlisted coordinates fixed. The <Weyl group> is therefore the group of signed permutations
$$
W\cong(\mathbb Z/2\mathbb Z)^n\rtimes S_n.
$$
Solved by gpt-5.6-sol high.
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