  We present an algorithmic proof of the Cartan-Dieudonn\'e theorem on
generalized real scalar product spaces with arbitrary signature. We use
Clifford algebras to compute the factorization of a given orthogonal
transformation as a product of reflections with respect to hyperplanes. The
relationship with the Cartan-Dieudonn\'e-Scherk theorem is also discussed in
relation to the minimum number of reflections required to decompose a given
orthogonal transformation.
