Normal forms for the G_2-action on the real symmetric 7x7-matrices by conjugation

The 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.Comment: 23 pages. Made typographical update in accordance with the final, published versio

Vector product algebras

Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.Comment: 7 page

The double sign of a real division algebra of finite dimension greater than one

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by a non-zero element are shown to form an invariant of A, called its double sign. The double sign causes the category of all real division algebras of a fixed dimension n>1 to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of the category of all finite-dimensional real division algebras.Comment: 12 page