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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Quaternion conjugation is an anti-involution

Statement

For the quaternion algebra $H$ of The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ , $\overset{ˉ}{\overset{ˉ}{a}} = a$ , $a + \overset{ˉ}{a} = \hat{2 a_{0}}$ for $a = (a_{0}, a_{1}, a_{2}, a_{3})$ , and $\overline{a b} = \overset{ˉ}{b} \overset{ˉ}{a}$ for all $a, b \in H$ .

Facts & Assumptions

Given: $a = (a_{0}, a_{1}, a_{2}, a_{3})$ and $b = (b_{0}, b_{1}, b_{2}, b_{3})$ in the real four-tuple algebra, with multiplication and conjugation as defined in The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ .

[F1]

Quaternion conjugation sends $(x_{0}, x_{1}, x_{2}, x_{3})$ to $(x_{0}, - x_{1}, - x_{2}, - x_{3})$ , and the four coordinates of multiplication are the displayed bilinear formula in The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ .

Proof

1.1givenF1algebra

Applying the defining conjugation formula twice returns $(a_{0}, a_{1}, a_{2}, a_{3})$ , so $\overset{ˉ}{\overset{ˉ}{a}} = a$ . Adding the tuple to its conjugate gives $(2 a_{0}, 0, 0, 0) = \hat{2 a_{0}}$ .

2.1givenF1step 1.1algebra∎

Write $a b = (r_{0}, r_{1}, r_{2}, r_{3})$ , where $r_{0} = a_{0} b_{0} - a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}$ , $r_{1} = a_{0} b_{1} + a_{1} b_{0} + a_{2} b_{3} - a_{3} b_{2}$ , $r_{2} = a_{0} b_{2} + a_{2} b_{0} + a_{3} b_{1} - a_{1} b_{3}$ , and $r_{3} = a_{0} b_{3} + a_{3} b_{0} + a_{1} b_{2} - a_{2} b_{1}$ . Substituting $(b_{0}, - b_{1}, - b_{2}, - b_{3})$ and $(a_{0}, - a_{1}, - a_{2}, - a_{3})$ into the same product formula gives $\overset{ˉ}{b} \overset{ˉ}{a} = (r_{0}, - r_{1}, - r_{2}, - r_{3})$ coordinate by coordinate. This is $\overline{a b}$ by [F1].

Depends on

The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$

Used by

Octonion conjugation is an anti-involution Lemma

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

N. Jacobson, Lectures on Quadratic Jordan Algebras, §8 (standard reference, not scraped)
John C. Baez, The Octonions, §2.3 (standard reference, not scraped)