  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.
