Definition:Cayley-Dickson Construction

Definition
Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

The Cayley-Dickson Construction on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows.

Let:
 * $A' = \struct {A'_F, \oplus'} = \struct {A, \oplus}^2$

where $\struct {A, \oplus}^2$ denotes the Cartesian product of $\struct {A, \oplus}$ with itself.

Then $\oplus'$ and $*'$ are defined on $A'$ as follows:
 * $\tuple {a, b} \oplus' \tuple {c, d} = \tuple {a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}$
 * ${\tuple {a, b}^*}' = \tuple {a^*, -b}$

where:
 * $\tuple {a, b}, \tuple {c, d} \in A'$
 * $a^*$ is the conjugation of $a \in A$.

If $\map \dim {A_F}$ is $d$, then $\map \dim {A'_F}$ is $2 d$.

Also see

 * Cayley-Dickson Construction forms $*$-Algebra