Definition:Octonion

Definition
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 $\left({a, b}\right) \in \Bbb O$, where:


 * $\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a \oplus d + c \oplus b}\right)$
 * $\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$

where:
 * $\overline a$ is the conjugate on $a$

and
 * $\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb O$.