The degree of an eight-dimensional real quadratic division algebra is 1, 3, or 5

A celebrated theorem of Hopf (1940) [11], , Bott and Milnor (1958) [1], , and Kervaire (1958) [12], states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified (Dieterich (2005) [6], , Dieterich (1998) [3], , Dieterich and Öhman (2002) [9], ), the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp. It was conjectured in Dieterich et al. (2006) [7].