  The fundamental tool in the classification of orthogonal coordinate systems
in which the Hamilton-Jacobi and other prominent equations can be solved by a
separation of variables are second order Killing tensors which satisfy the
Nijenhuis integrability conditions. The latter are a system of three non-linear
partial differential equations. We give a simple and completely algebraic proof
that for a Killing tensor the third and most complicated of these equations is
redundant. This considerably simplifies the classification of orthogonal
separation coordinates on arbitrary (pseudo-)Riemannian manifolds.
