Proposition 5

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

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

*Let $H$ and $H'$ be two parallel, non-isotropic hyperplanes of $E$ . There exists a unique vector $v \in V$ orthogonal to $H$ and such that $H' = H + v$ . The displacement $s_{H'} s_H$ is the translation by the vector $2v$ .

The existence and uniqueness of $v$ are immediate. The automorphism of $V$ associated to $s_{H'} s_H$ is the identity; thus $s_{H'} s_H$ is a translation. On the other hand, let $a \in H'$ ; then $a - v \in H$ and

s_{H'} s_H(a - v) = s_{H'}(a - v) = a + v = (a - v) + 2v,

showing that $s_{H'} s_H$ is the translation by the vector $2v$ .

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

Cited by

Remark, § 2
Proposition 3, § 3
Proposition 9, § 3
Proposition 8, § 2

Cites

Nothing.