Cayley-Dickson Construction forms Star-Algebra

Theorem
Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

Let $A' = \struct {A'_F, \oplus'} = \struct {A, \oplus}^2$ 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 $\tuple {a_1, b_1}, \tuple {a_2, b_2}, \tuple {c, d} \in A'$.

Then:

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

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

Then:

Similarly:
 * $\tuple {a, b} \oplus' \paren {\tuple {c, d} \beta} = \tuple {a, b} \oplus' \tuple {c, d} \beta$

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

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

So:

Finally:

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

Hence the result.