Nicely Normed Alternative Algebra is Normed Division Algebra

Theorem
$A = \struct {A_F, \oplus}$ be a nicely normed $*$-algebra which is also an alternative algebra.

Then $A$ is a normed division algebra.

Proof
Let $a, b \in A$.

Then all of $a, b, a^*, b^*$ can be generated by $\map \Im a$ and $\map \Im b$.

So as $A$ is an alternative algebra, it follows that $\oplus$ is associative for $a, b, a^*, b^*$.

So: