Integrable geodesic flows with simultaneously diagonalisable quadratic integrals

We show that if $n$ functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an $n$-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the St\"ackel construction, so the metric admits orthogonal separation of variables.