\begin{definition}[Definition:Cayley-Dickson Construction]
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.
The '''Cayley-Dickson Construction''' on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows.
Let:
: $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$
where $\left({A, \oplus}\right)^2$ denotes the Cartesian product of $\left({A, \oplus}\right)$ with itself.
Then $\oplus'$ and $*'$ are defined on $A'$ as follows:
:$\left({a, b}\right) \oplus' \left({c, d}\right) = \left({a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\right)$
:${\left({a, b}\right)^*}' = \left({a^*, -b}\right)$
where:
: $\left({a, b}\right), \left({c, d}\right) \in A'$
: $a^*$ is the conjugation of $a \in A$.
{{ExtractTheorem}}
If $\dim \left({A_F}\right)$ is $d$, then $\dim \left({A'_F}\right)$ is $2 d$.
\end{definition}