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.

Symmetrized product of an associative algebra

Definition

Let $k$ be a field with $char (k) \neq 2$ and let $A$ be a unital associative $k$ -algebra as in Algebras over a commutative ring, central structure maps, and algebra homomorphisms. On the underlying $k$ -vector space define

$a \circ b : = \frac{1}{2} (a b + b a) .$

Denote this algebra with product by $A^{+}$ . The product is symmetric and bilinear. This definition alone does not assert the Jordan identity; the following theorem proves it. It does not assert that a Jordan product is associative.

Depends on

Used by

The symmetrized product need not be associative Counterexample
The spin factor and the plus-quadratic Clifford convention Proposition
The symmetrized product of an associative algebra is Jordan 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

N. Jacobson, Lectures on Quadratic Jordan Algebras, §1 (standard reference, not scraped)
John C. Baez, The Octonions, §3.2, PDF p. 20 (standard reference, not scraped)