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.

Hermitian three-by-three octonion matrix data

Definition

Let

$H_{3} (O) : = {(\begin{array}{ccc} a & x & \overset{ˉ}{z} \\ \overset{ˉ}{x} & b & y \\ z & \overset{ˉ}{y} & c \end{array}) : a, b, c \in R, x, y, z \in O} .$

For octonion matrices define $(X Y)_{i j} : = \sum_{r = 1}^{3} X_{i r} Y_{r j}$ , with each octonion product taken in the displayed order. Define the ambient symmetrized expression

$m (X, Y) : = \frac{1}{2} (X Y + Y X) \in M_{3} (O) .$

Here $M_{3} (O)$ is only the real vector space of $3 \times 3$ octonion matrices with this specified binary matrix product; associativity is not assumed. The theorem The real Albert algebra is a Jordan algebra ↗ proves that $m (X, Y)$ is Hermitian for $X, Y \in H_{3} (O)$ and satisfies the Jordan identity. No claim is made for other matrix sizes.

Depends on

Used by

An off-diagonal product in the Albert algebra Example
The real Albert algebra is a Jordan algebra Theorem

Dependency tree · two levels

10 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, §§6.4–6.7 (standard reference, not scraped)