Let M be a closed smooth Spin 7-manifold. The peculiar features of the Hopf-invariant 1 dimensions mean that the linking form of M admits a quadratic refinement q which is not always homogeneous. In my PhD thesis I defined q using a co-bounding Spin 8-manifold for M. However the recently constructed 7-manifolds, P_k, of Grove, Verdiani and Ziller have made it important to find an intrinsic definition of q. In this talk I will give such a definition of q using the index theory of Dirac operators twisted by quaternionic line bundles. The main application is Goette's recent diffeomorphism classification of the manifolds P_k. As time permits I will mention other points of interest: not all of them in dimension 7. This work is joint with S. Goette.