ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A concrete octonion associator witness

Statement

In the pair convention for $O$ , the associator $[i, j, ℓ] : = (i j) ℓ - i (j ℓ)$ is $2 k ℓ \neq 0$ . Both alternative laws still hold, so this is a concrete alternative but nonassociative algebra.

Facts & Assumptions

Given: The named basis units and multiplication on $O$ .

[F1]

The pair formula gives $i j = k$ , $(0, k) = k ℓ$ , and $i (j ℓ) = (0, j i)$ with $j i = - k$ (Real octonions by Cayley–Dickson doubling, The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ ).

[F2]

The same algebra satisfies both alternative laws (The Cayley–Dickson octonions are alternative but not associative).

Proof

1.1givenF1algebra

From [F1], $(i j) ℓ = (0, k) = k ℓ$ and $i (j ℓ) = (0, - k) = - k ℓ$ . Thus $[i, j, ℓ] = k ℓ - (- k ℓ) = 2 k ℓ$ . This is nonzero because $k \neq 0$ in $H$ and the real scalar $2$ is nonzero.

2.1step 1.1F2∎

By [F2], the left and right alternative laws hold for every pair of octonions, while step 1.1 gives a nonzero associator. Hence this example is alternative and not associative.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)
Holger P. Petersson, Albert Algebras, §3.5 (standard reference, not scraped)