Definition:Octonion

Definition

The set of octonions, usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\H$ as follows:

Let $a, b \in \H$.

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 $\Bbb O$.

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:

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

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.

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.

1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cayley algebra (J.T. Graves, 1843)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): octonion