Integral geometry of exceptional spheres

The algebras of valuations on S⁶and S⁷invariant under the actions of G₂and Spin(7) are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in S⁶and S⁷is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations.