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:

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

Proof

\(\ds \overline {\paren {\overline {\mathbf x} } }\)

\(=\)

\(\ds \overline {\paren {\overline {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} } }\)

Definition of $\mathbf x$

\(\ds \)

\(=\)

\(\ds \overline {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}\)

Definition of Quaternion Conjugate

\(\ds \)

\(=\)

\(\ds a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k\)

Definition of Quaternion Conjugate

\(\ds \)

\(=\)

\(\ds \mathbf x\)

Definition of $\mathbf x$

$\blacksquare$