 Cartan–Dieudonné theorem

Every orthogonal transformation of a finite-dimensional nondegenerate quadratic space over a field of characteristic different from two is a product of at most its dimension many hyperplane reflections. In the real positive-definite case, a reflection normal to $gv-v$ sends $gv$ to $v$ whenever $gv\ne v$. Composing with that reflection fixes $v$ and leaves an orthogonal transformation of $v^{\perp }$, so induction proves the bound. Each reflection has determinant minus one, hence orientation-preserving transformations use an even number of reflections.

The Cartan–Dieudonné theorem is a result in differential geometry and linear algebra that characterizes elements of a projective space using linear combinations of certain vectors. Specifically, it is often described in the context of the geometry of vector spaces and the projective spaces constructed from them.