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 $\Bbb O$ cannot be a associative algebra.

Proof of Normed Division Algebra

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$\blacksquare$

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