Corollary

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

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

If V is finite dimensional, every reflection is of determinant −1.

Let s be a reflection in V. Prop. 2, (i) shows that there exists a basis $(e_1, \ldots, e_n)$ of V such that $s(e_1) = e_1, \ldots, s(e_{n-1}) = e_{n-1}$ and $s(e_n) = -e_n$ , hence $\det(s) = -1$ .

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

Cited by

Nothing.

Cites

Prop. 2