On the dimension of a composition algebra (Q1919363)

In this note on composition algebras and the associated anticommutative algebras of trace zero elements (called vector product algebras by the author) a relation is derived which, if the underlying field has characteristic zero, yields the familiar result that the dimension of a composition algebra equals 1, 2, 4 or 8. The proof is based on identities of vector product algebras (including the restriction of the norm form), and uses a particular basis after passing to the algebraic closure of the underlying field. Although this proof is quite ingenious and surprising, it seems that the basic ingredients (identities and properties of the norm form) are not fundamentally different from those used in the classical proof. (It would be interesting to learn more about the ``graphical technique'' mentioned by the author in the introduction).