Proposition 6

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

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

Let s be the orthogonal reflection with respect to a line D of V. If $\Delta$ and $\Delta'$ are two half-lines starting at the origin (resp. two lines passing through the origin) of V, we have

(s(\Delta), \widehat{s(\Delta')}) \equiv - (\widehat{\Delta}, \widehat{\Delta'}) \pmod{2\pi} \quad \text{(resp. } \pmod{\pi})).

Let $u$ be a rotation transforming $\Delta$ into $\Delta'$ . Since $su$ is an orthogonal transformation of V of determinant $-1$ , it is a reflection and thus $(su)^2 = 1$ . Consequently, $u^{-1} = sus^{-1}$ transforms $s(\Delta)$ into $s(\Delta')$ , hence the proposition.

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

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