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.