Cayley-Dickson construction

Definition
Suppose $$R$$ is a nonzero commutative unital ring and $$A$$ is a normed algebra over $$R$$, i.e., a unital non-associative (in the sense of not necessarily associative) algebra over $$R$$ with a norm function to $$R$$ (a nondegenerate quadratic form that is a multiplicative homomorphism).

Suppose there is an involution $${}^*$$ for $$A$$ over $$R$$. This means that $$(ab)^* = b^*a^*$$ and $$(a^*)^* = a$$ for all $$a,b \in A$$, $${}^*$$ is $$R$$-linear, and it preserves the norm function.

Suppose $$\lambda$$ is an invertible element of $$R$$. Then, we can define a new algebra $$KD(A,\lambda)$$ over $$A$$ as follows:


 * As a $$R$$-module, $$KD(A,\lambda) = A \oplus A$$.
 * The multiplication is defined by $$(a \oplus b)(c \oplus d) = (ac + \lambda d^*b) \oplus (da + bc^*)$$
 * The extended involution is defined as $$(a \oplus b)^* = a^* \oplus (-b)$$
 * The extended norm is defined as $$N(a \oplus b) = (a \oplus b)(a \oplus b)^*$$.

This construction is termed the Cayley-Dickson construction and algebras constructed in this way are termed Cayley-Dickson algebras.

Facts

 * Bruck-Kleinfeld theorem on alternative division rings tells us that any alternative division ring that is not associative must be a Cayley-Dickson algebra.

Particular cases

 * Quaternion algebra over a field is a four-dimensional algebra obtained by applying the Cayley-Dickson construction twice to a field (both applications treat the original field as the ground ring).
 * Cayley octonion algebra over a field is an eight-dimensional algebra obtained by applying the Cayley-Dickson construction thrice to a field (all three applications treat the original field as the ground ring).