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 Stäckel construction, so the metric admits orthogonal separation of variables.