# Octonions form Algebra

## Theorem

The set of octonions $\Bbb O$ forms an algebra over the field of real numbers.

This algebra is:

- $(1): \quad$ An alternative algebra, but
**not**an associative algebra. - $(2): \quad$ A normed division algebra.
- $(3): \quad$ A nicely normed $*$-algebra.

## Proof

The octonions $\Bbb O$ are formed by the Cayley-Dickson construction from the quaternions $\Bbb H$.

From Quaternions form Algebra, we have that $\Bbb H$ forms:

- $(1): \quad$ An associative algebra
- $(2): \quad$ A normed division algebra
- $(3): \quad$ A nicely normed $*$-algebra.

From Cayley-Dickson Construction forms Star-Algebra, $\Bbb O$ is a $*$-algebra.

From Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed, $\Bbb O$ is a nicely normed $*$-algebra.

From Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative, $\Bbb O$ is an alternative algebra.

Now suppose $\Bbb O$ formed an associative algebra.

Then from Cayley-Dickson Construction from Commutative Associative Algebra is Associative, that would mean $\Bbb H$ is a commutative algebra.

But from Quaternions form Algebra it is explicitly demonstrated that $\Bbb H$ is **not** a commutative algebra.

So $\Bbb O$ can not be a associative algebra.

### Proof of Normed Division Algebra

$\blacksquare$

## Sources

- 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