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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:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$


$\overline a$ is the conjugate on $a$


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

Octonion Addition

Let $x = \tuple {a, b}$ and $y = \tuple {c, d}$ be octonions, where $a, b, c, d \in \H$ are quaternions.

The sum of $x$ and $y$ is defined as:

$x + y = \tuple {a, b} + \tuple {c, d} = \tuple {a + c, b + d}$

Also known as

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

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

Also see

  • Results about octonions can be found here.

Historical Note

The octonions were discovered by John Thomas Graves in December $1843$, in the wake of the work he did with William Rowan Hamilton on the quaternions.

However, he never made it into print, and it was Arthur Cayley who published his own work on octonions in $1845$, hence claiming the credit.

Graves' precedent was discovered in $1847$ on the evidence of a letter he had written to Hamilton on the occasion of his initial discovery of them.