\begin{definition}[Definition:Octonion]
The set of '''octonions''', usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\Bbb H$ as follows:
From Quaternions form Algebra, $\Bbb H$ forms a nicely normed $*$-algebra.
Let $a, b \in \Bbb H$.
Then $\tuple {a, b} \in \Bbb O$, where:
:$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
:$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$
where:
:$\overline a$ is the conjugate on $a$
and 
:$\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.
\end{definition}