DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Real spin-factor product

Definition

Let $(V, ⟨ \cdot, \cdot ⟩)$ be a real inner-product space with the positive-definite symmetric form of Real and complex inner product spaces, with the inner product linear in the first argument. On $S (V) : = V \oplus R$ define

$(v, α) \circ (w, β) : = (α w + β v, ⟨ v, w ⟩ + α β) .$

This is the real spin-factor product; The real spin factor satisfies the Jordan identity ↗ proves that it is a unital Jordan algebra.

Depends on

Used by

The spin factor on the real plane Example
The spin factor and the plus-quadratic Clifford convention Proposition
The real spin factor satisfies the Jordan identity Theorem

Dependency tree · two levels

6 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

John C. Baez, The Octonions, §3.4, PDF pp. 27–28 (standard reference, not scraped)
Holger P. Petersson, Albert Algebras, §4.1 (standard reference, not scraped)