\begin{definition}[Definition:Quaternion/Construction from Cayley-Dickson Construction]
The set of quaternions $\Bbb H$ can be defined by the Cayley-Dickson construction from the set of complex numbers $\C$.
From Complex Numbers form Algebra, $\C$ forms a nicely normed $*$-algebra.
Let $a, b \in \C$.
Then $\left({a, b}\right) \in \Bbb H$, where:
:$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$
:$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$
where:
:$\overline a$ is the complex conjugate of $a$
and 
:$\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb H$.
It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions.
\end{definition}