Definition:Quaternion/Construction from Cayley-Dickson Construction

Definition

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 $\tuple {a, b} \in \Bbb H$, 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 complex conjugate of $a$

and

$\overline {\tuple {a, b} }$ 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.

Sources

• John C. Baez: The Octonions (2002): 1 Introduction