Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields (Q6633599)

Killing tensor fields on the space of constant curvature and on the complex projective space can be decomposed into sums of symmetric tensor products of Killing vector fields. Motivated by this fact, the authors investigate whether this property holds for Killing tensor fields on all Riemannian symmetric spaces. The question is settled in the negative by providing examples of quadratic Killing tensors which are not quadratic forms in the Killing vector fields on the quaternionic projective space \(\mathbb HP^n\), \(n \geq 3\), and on the Cayley projective plane \(\mathbb OP^2\).