Ternary composition algebras and Hurwitz’s theorem

Abstract

Axioms are proposed for a certain ‘‘alternative’’ kind of ternary composition algebra, termed a 3Cn algebra. The axioms are shown to be (for n>2) in a simple correspondence with the axioms for a ternary vector cross product algebra. The axioms imply that n=1, 2, 4, or 8 (from which the usual Hurwitz theorem is deduced). The existence of 3C8 algebras is demonstrated by an explicit construction in four-dimensional Hilbert space, without appeal to the properties of the algebra of octonions.