Cayley-Dickson Construction from Real Star-Algebra is Commutative

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 a real star-algebra $A'$ is a commutative algebra.

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

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

Let $A$ be a real star-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.