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 $\tuple {a, b} \in \Bbb O$, 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 of $a$

and
 * $\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.

Also known as
The octonions are sometimes referred to as the Cayley numbers, for.

Some sources report them as the Graves-Cayley Numbers, for, who actually discovered them.