CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Connected nilpotent Lie groups are central quotients of BCH groups

Statement

Assume countable choice. Every connected real Lie group with nilpotent Lie algebra is a quotient of the BCH group on that algebra by a discrete central subgroup.

Facts & Assumptions

Given: Countable choice and a connected group $G$ with nilpotent Lie algebra $n$ .

[L1]

The simply connected cover of $G$ is quotiented by a discrete central kernel (Connected Lie groups are central quotients of simply connected integrations).

[L2]

A simply connected nilpotent integration is isomorphic through its exponential to the BCH group (Exponential diffeomorphism for simply connected nilpotent Lie groups).

Proof

technique · combine the two classifications

1.1

By [L1], $G ≅ \tilde{G} / Γ$ for a discrete central subgroup of its simply connected cover. The cover has the same nilpotent Lie algebra.

L1given

2.1

By [L2], exponential coordinates identify $\tilde{G}$ with the BCH group on $n$ . Transporting $Γ$ through this isomorphism preserves discreteness and centrality and gives the asserted quotient. The trivial kernel and zero-dimensional group are included. The countable-choice use is exactly that already declared in [L1]–[L2].

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Knapp, Lie Groups Beyond an Introduction, Theorem 1.127 and Corollary 1.134 (standard reference, not scraped)