# 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.

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

$\blacksquare$