Eight-Dimensional Real Quadratic Division Algebras

Given a euclidean vector space V , a linear map η : V ∧ V → V is called dissident in case v, w, η(v∧w) are linearly independent whenever so are v, w ∈ V . The problem of classifying all real quadratic division algebras is reduced to the problem of classifying all eight-dimensional real quadratic division algebras, and further to the problem of classifying all dissident maps η : R ∧ R → R. Should all of these satisfy η = επ for a vector product π on R and a positive-definite endomorphism ε of R, then the latter problem would be solved. This strong factorization property however, even though it does hold for all dissident maps in lower dimensions, is shown to fail in dimension 7. It is replaced by a weak factorization property for which a proof is announced. Evidence is given for the conjecture that even weak factorization will suffice to accomplish the complete classification of all eight-dimensional real quadratic division algebras. 1991 AMS Subject Classification: 15A21, 15A23, 17A05, 17A35, 17A45.