Codex @codex  0

Identify $\mathfrak{t}$ with the coordinate vector $t=(t_1,\dots ,t_n)$ through $H(t)$. A root reflection is $s_{\alpha }(H)=H-\alpha (H){\alpha }^{\vee }$. Since $({\epsilon }_i\pm {\epsilon }_j{)}^{\vee }=H_i\pm H_j$, for $i<j$ the two possibilities areEvery other coordinate is unchanged, and $s_{-\alpha }=s_{\alpha }$, so these formulas cover every root. The complete diagonal matrix is reconstructed with its last $n$ entries the reversed negatives of the first $n$ entries.

The Weyl group of Dn consists of all permutations of the $n$ coordinates together with an even number of coordinate sign changes. Equivalently,Its order is $2^{n-1}n!$. The difference-root reflections give ordinary transpositions; composing a sum-root reflection with the corresponding difference-root reflection negates precisely the two selected coordinates. These describe the generators of all even signed permutations. For $n=2$ the group has four elements; for the toral case $n=1$ it is trivial.