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 an element a ∈ A ∖{0} are shown to form an invariant of A , called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {D}_n$\end{document} of all n ‐dimensional real division algebras 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 \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {D}_n$\end{document} .