Let $W$ be the Weyl group corresponding to the maximal torus $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ in a Borel group of $\operatorname{SO}_{2n}(\mathbb C)$.

What are the matrices corresponding to simple reflections of $W$?

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I believe an answer to your question is in the Bourbaki volume dedicated to groups generated by reflections, Lie ..., vol. VI...

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unlessone takes a particular quadratic form (in which case itisBorel), right? (I suppose one could say equivalently that the group of upper-triangular matrices isn't Borel unless one chooses a basis adapted appropriately to the choice of quadratic form.) $\endgroup$1more comment