# 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

 $\displaystyle \overline {\paren {\overline {\mathbf x} } }$ $=$ $\displaystyle \overline {\paren {\overline {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} } }$ 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 $\mathbf x$

$\blacksquare$