Remark

Tag 0APP, permanent. Cite it as [Bourbaki, Tag 0APP].

Lie Groups and Lie Algebras, chapter V, § 2. Reflections

Let $a \neq 0$ be in V and let $a^*$ be a non-zero linear form on V: it follows from formula (1) that

s_{a,a^*}^2(x) = x + (\langle a, a^* \rangle - 2)\langle x, a^* \rangle.a

and hence that $s_{a,a^*}$ is a reflection if and only if $\langle a, a^* \rangle = 2$ . In this case, we have $s_{a,a^*}(a) = -a$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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