= Cartan–Dieudonné theorem
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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.
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