Octonion Conjugation is Involution

Theorem
Let $x = \left({a, b}\right): a, b \in \mathbb H$ be a octonion.

Let $\overline x$ be the conjugate of $x$.

Then:
 * $\overline \cdot : x \mapsto \overline x$

is an involution.

That is:
 * $\overline{\left({\overline x}\right)} = x$