Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative

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 alternative algebra $A$ is a nicely normed associative 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:

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:
 * $\paren {\tuple {a, b} \tuple {a, b} } \tuple {c, d} = \tuple {a, b} \paren {\tuple {a, b} \tuple {c, d} }$

Similarly it can be shown that:
 * $\paren {\tuple {c, d} \tuple {a, b} } \tuple {a, b} = \tuple {c, d} \paren {\tuple {a, b} \tuple {a, b} }$

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

It follows from reversing the chain of equalities that if $A'$ is a nicely normed and alternative algebra then $A$ has to 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 $A$ is also a nicely normed algebra.

Hence the result.