Cayley–Dickson construction

Summary

The Cayley–Dickson construction doubles an algebra with an involution to an algebra of twice the dimension with a new involution. Starting from $\R$, it produces in turn $\C$, the quaternions, the octonions and the sedenions. From $\C$ on, each step loses a property. Commutativity is lost at the quaternions, associativity at the octonions, and the alternative law and division at the sedenions. With a parameter in the field it also produces the split forms. Over a field of characteristic not $2$ it produces all unital composition algebras, by Hurwitz's theorem.