Definition:Quaternion/Construction from Cayley-Dickson Construction

Definition
The quaternions 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 $a, b \in \Bbb H$, 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.