Third power associative composition algebras

Classical composition algebras, with a unit element, are well-known and can be obtained by means of the Cayley-Dickson doubling process. If the condition on the existence of unit element is dropped, many new algebras arise. However, it is shown in this paper that if such a weak condition as the associativity of third powers of any element is imposed, only the known flexible composition algebras appear. Partially supported by the DGICYT (PS 90-0129) and by the DGA (PCB-6/91) Supported by a grant from the ‘Plan de Formación del Personal Investigador’ (DGICYT, Spain) This article was processed by the authors using the Springer-Verlag TEX P Jour1g macro package 1991.