Quadratic Killing tensors on classical Lie groups are decomposable

A Killing tensor field on a Riemannian manifold $(M,g)$ is a covariant symmetric tensor field whose contraction with the velocity vector along a geodesic produces a homogeneous polynomial first integral of the geodesic flow. Such a tensor is called \emph{decomposable} if it lies in the subalgebra generated by Killing vector fields; equivalently, the corresponding polynomial integral is then a polynomial in the linear integrals coming from infinitesimal isometries. On spaces of constant sectional curvature and on the complex projective space, every Killing tensor field is decomposable. By contrast, the quaternionic projective spaces and the Cayley projective plane admit indecomposable quadratic Killing tensor fields. We prove that every quadratic Killing tensor field on the compact classical Lie groups $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, equipped with a bi-invariant Riemannian metric, is decomposable; equivalently, every quadratic first integral of the geodesic flow on these groups is a quadratic polynomial in the linear first integrals.