Octonion Conjugation is Involution

Theorem
Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a octonion.

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

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

is an involution.

That is:
 * $\overline {\paren {\overline x} } = x$