Quaternion Conjugation is Involution

Theorem
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\overline {\mathbf x}$ denote the quaternion conjugate of $\mathbf x$.

Then the operation of quaternion conjugation is an involution:


 * $\overline {\paren {\overline {\mathbf x} } } = \mathbf x$