Quaternion Conjugation is Involution

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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

\(\displaystyle \overline {\paren {\overline {\mathbf x} } }\) \(=\) \(\displaystyle \overline {\paren {\overline {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} } }\) Definition of $\mathbf x$
\(\displaystyle \) \(=\) \(\displaystyle \overline {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}\) Definition of Quaternion Conjugate
\(\displaystyle \) \(=\) \(\displaystyle a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k\) Definition of Quaternion Conjugate
\(\displaystyle \) \(=\) \(\displaystyle \mathbf x\) Definition of $\mathbf x$

$\blacksquare$