Cayley-Dickson Construction from Commutative Associative Algebra is Associative

Theorem
Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

Let $A' = \struct {A_F, \oplus'}$ be constructed from $A$ using the Cayley-Dickson construction.

Then:
 * $A'$ is an associative algebra


 * $A$ is both a commutative algebra and an associative algebra.
 * $A$ is both a commutative algebra and an associative algebra.

Proof
Let the conjugation operator on $A$ be $*$.

Let $\tuple {a, b}, \tuple {c, d}, \tuple {e, f} \in A'$.

In order to streamline notation, let $\oplus$ and $\oplus'$ both be denoted by product notation:
 * $a \oplus b =: a b$
 * $x \oplus' y =: x y$

The context will make it clear which is meant.

Suppose $A$ is commutative and associative.

Then:

Now suppose $A'$ is an associative algebra.

By picking apart the above series of equations, it is clear that $A$ has to be both commutative and associative in order to assure the associativity of $A'$.