manuscripta math. 103, 75 – 90 (2000) c  Springer-Verlag 2000 Jos´ e Antonio Cuenca · Alberto Elduque · Jos´ e Mar´ ıa P´ erez-Izquierdo Power associative composition algebras Received: 1 February 2000 Abstract. Composition algebras of arbitrary dimension over a field and satisfying the iden- tities x 2 x = xx 2 and (x 2 ) 2 = (x 2 x)x are shown to be precisely the well-known unital composition algebras, with the exception of three two dimensional algebras over the field of two elements. 1. Introduction and main result A nonassociative (i.e. not necessarily associative) algebra over a field F is said to be a composition algebra if it is equipped with a nondegenerate quadratic form (the norm) n : A -→ F such that n(xy) = n(x)n(y) (1) for any x,y ∈ A. The form being nondegenerate means that if the associated bilinear form is given by n(x,y) = n(x + y) - n(x) - n(y), then {x ∈ A : n(x) = n(x, A) = 0}= 0. The norm is said to be strictly nondegenerate if A ⊥ ={x ∈ A : n(x, A) = 0}= 0. In case the characteristic of F is not two, n(x) = 1 2 n(x,x) and both concepts agree. J. A. Cuenca: Departamento de Algebra, Geometr´ ıa y Topolog´ ıa, Universidad de M´ alaga, 29080 M´ alaga, Spain A. Elduque: Departamento de Matem´ aticas y Computaci ´ on, Universidad de La Rioja, 26004 Logro˜ no, Spain. Present address: Departamento de Matem´ aticas, Universidad de Zaragoza, 50009 Zaragoza, Spain. e-mail: elduque@posta.unizar.es J. M. P´ erez-Izquierdo: Departamento de Matem´ aticas, Universidad de Zaragoza, 50009 Zaragoza, Spain. Present address: Departamento de Matem´ aticas y Computaci´ on, Universidad de La Rioja, 26004 Logro ˜ no, Spain Mathematics Subject Classification (2000): Primary 17A75; Secondary 17A05