Killing forms on symmetric spaces

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p  -form (p⩾2p⩾2) if and only if it isometric to a Riemannian product Sk×NSk×N, where SkSk is a round sphere and k>pk>p.