Octonion Conjugation is Involution

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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:

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


Proof

\(\ds \overline {\paren {\overline x} }\) \(=\) \(\ds \overline {\paren {\overline {\tuple {a, b} } } }\) Definition of Octonion
\(\ds \) \(=\) \(\ds \overline {\tuple {\overline a, -b} }\) Definition of Conjugate of Octonion
\(\ds \) \(=\) \(\ds \tuple {\overline {\paren {\overline a} }, -\paren {-b} }\) Definition of Conjugate of Octonion
\(\ds \) \(=\) \(\ds \tuple {a, b}\) Quaternion Conjugation is Involution
\(\ds \) \(=\) \(\ds x\) Definition of Octonion

$\blacksquare$