Normal forms for the -action on the real symmetric 7×7-matrices by conjugation

The exceptional Lie group acts on the set of real symmetric 7×7-matrices by conjugation. We solve the normal form problem for this group action. In view of the earlier results [G.M. Benkart, D.J. Britten, J.M. Osborn, Real flexible division algebras, Canad. J. Math. 34 (1982) 550–588; J.A. Cuenca Mira, R. De Los Santos Villodres, A. Kaidi, A. Rochdi, Real quadratic flexible division algebras, Linear Algebra Appl. 290 (1999) 1–22; E. Darpö, On the classification of the real flexible division algebras, Colloq. Math. 105 (1) (2006) 1–17], this gives rise to a classification of all finite-dimensional real flexible division algebras. By a classification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of all real symmetric matrices, based on eigenvalues.