Cayley-Dickson Construction from Real Star-Algebra is Commutative

Theorem
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.

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

Then $A$ is a real algebra iff $A'$ is a commutative algebra.

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

Let $\left({a, b}\right), \left({c, d}\right) \in A'$.

Let $A$ be a real algebra.

So $A'$ is a commutative algebra.

Let $A'$ be a commutative algebra.

By picking apart the above equations, it is clear that for $A'$ to be a commutative algebra it is necessary for $A$ to be both real and commutative.

Hence the result.