Killing forms on G2G2- and Spin7Spin7-manifolds

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We prove that on a compact manifold with holonomy G2G2 or Spin7Spin7 any Killing form has to be parallel. The main tool is a universal Weitzenböck formula. We show, how such a formula can be obtained for any given holonomy group and any representation defining a vector bundle.