2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131402The exceptional Lie group G_2 acts on the set of real symmetric 7x7-matrices by conjugation. We solve the normal form problem for this group action. In view of earlier results, 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.23 pages. Made typographical update in accordance with the final, published versionRings and Algebras17A35text