Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative

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 nicely normed alternative algebra iff $A$ is a nicely normed associative algebra.

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

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

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 associative algebra.

Then:

Similarly:

Thus it can be seen that:
 * $\left({\left({a, b}\right) \left({a, b}\right)}\right) \left({c, d}\right) = \left({a, b}\right) \left({\left({a, b}\right) \left({c, d}\right)}\right)$

Similarly it can be shown that:
 * $\left({\left({c, d}\right) \left({a, b}\right)}\right) \left({a, b}\right) = \left({c, d}\right) \left({\left({a, b}\right) \left({a, b}\right)}\right)$

and so $A'$ is seen to be an alternative algebra.

It follows from reversing the argument that if $A'$ is not a nicely normed and alternative algebra then $A$ will not be a nicely normed associative algebra.

Then from Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed, we have that $A'$ is a nicely normed algebra iff $A$ is also a nicely normed algebra.

Hence the result.