Produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one; named after Arthur Cayley and Leonard Eugene Dickson.
Repeatedly applied, it loses first order, then commutativity of multiplication, then associativity, and finally alternativity.
More generally, it takes any algebra with involution to another algebra with involution of twice the dimension; from the sedenions on, the algebras have nontrivial zero divisors.