Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed

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 nicely normed algebra $A$ is also a nicely normed algebra.

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

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

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.

Let $A$ be a nicely normed algebra.

Then:

As $A$ is itself nicely normed, we have that $a + a^*$ is real.

Hence it follows that $\tuple {a + a^*, 0}$ is real.

Next:

We have by definition of norm that $\norm a^2 + \norm b^2$ is real.

Hence it follows that $\tuple {\norm a^2 + \norm b^2, 0}$ is real.

It follows from reversing the argument that if $A'$ is not nicely normed then nor will $A$ be.