It is wellknown the Hurwitz theorem showing that every unital composition algebra is of dimension 1, 2, 4 or 8 over the base field.
Is there an analogue in the non unital case? ¿Do non unital composition algebras of arbitrary dimension exist?
It is wellknown the Hurwitz theorem showing that every unital composition algebra is of dimension 1, 2, 4 or 8 over the base field. Is there an analogue in the non unital case? ¿Do non unital composition algebras of arbitrary dimension exist? 


If $A$ is a finitedimensional algebra over a field, with a strictly nondegenerate multiplicative quadratic form, then the dimension of $A$ is 1, 2, 4 or 8. This result can be found in "The Book of Involutions" by Knus, Merkurjev, Rost and Tignol (Amer. Math. Soc. Coll. Publ. 44, Providence, 1998), Corollary (33.28). The argument for this is due to Kaplansky (Proc. Amer. Math. Soc 4, 1953), and consists of defining a new multiplication on $A$ which admits a unit element. 

