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A celebrated theorem of Hopf, Bott, Milnor, and Kervaire states that everyfinite-dimensional real division algebra has dimension 1, 2, 4, or 8. While thereal division algebras of dimension 1 or 2 and the real quadratic divisionalgebras of dimension 4 have been classified, the problem of classifying all8-dimensional real quadratic division algebras is still open. We contribute toa solution of that problem by proving that every 8-dimensional real quadraticdivision algebra has degree 1, 3, or 5. This statement is sharp.

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