Cayley-Dickson Construction forms Star-Algebra

Theorem
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.

Let $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$ be the algebra formed from $A$ by the Cayley-Dickson construction.

Then $A'$ is also a $*$-algebra.

Bilinearity of $\oplus'$
First we need to show that $\oplus'$ is bilinear.

$(1): \quad$ Let $\left({a_1, b_1}\right), \left({a_2, b_2}\right), \left({c, d}\right) \in A'$.

Then:

Similarly (and equally tediously) we can show that:
 * $\left({c, d}\right) \oplus' \left({\left({a_1, b_1}\right) + \left({a_2, b_2}\right)}\right) = \left({\left({c, d}\right) \oplus' \left({a_1, b_1}\right)}\right) + \left({\left({c, d}\right) \oplus' \left({a_2, b_2}\right)}\right)$

$(2): \quad$ Let $\left({a, b}\right), \left({c, d}\right) \in A'$ and $\alpha, \beta \in \R$.

Then:

Similarly, $\left({a, b}\right) \oplus' \left({\left({c, d}\right) \beta}\right) = \left({a, b}\right) \oplus' \left({c, d}\right) \beta$.

So $\oplus'$ has been shown to be a bilinear mapping.

Conjugate Nature of $*'$
We have that:
 * $\forall \left({a, b}\right) \in A': {\left({a, b}\right)^*}' = \left({a^*, -b}\right)$

So:

Finally:

thus proving that $*'$ is a conjugation on $A'$.

Hence the result.