Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]
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The constructions in Definition [fcap-001R]--Example [fcap-001V] start with a smooth homomorphism. Their tangent map, bracket law, functor laws, and exponential naturality therefore do not require a theorem that upgrades a continuous homomorphism to a smooth one. The generic theorem does not invoke the closed-subgroup theorem; that theorem belongs to the later specialization to concrete matrix subgroups. This is the dependency split proposed in TauCetiRoadmap PR 224.

The exponential input is separate from the tangent and bracket input. Likewise, the Baker--Campbell--Hausdorff calculation can first be made in a matrix or general-linear group; using it inside a particular closed matrix subgroup additionally requires the subgroup's Lie structure. Liu describes the local exponential and BCH coordinates, while Isaev--Rubakov give the matrix Campbell--Hausdorff calculation.

The open TauCeti draft PR 3078 proposes the generic construction and its functor and exponential laws. It is not merged, so these cards carry no Lean markers. Even after that generic interface lands, the differential of \(\operatorname {Spin}(Q)\to SO(Q)\) still needs compatible Lie-group structures on the two abstract groups. The algebraic identification in Theorem [fcap-000R] does not by itself supply those structures.

Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]
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Reference. Theory of groups and symmetries: finite groups, Lie groups, and Lie algebras [isaev2018theory]

Reference. Notes for lie groups and representations [liu2016lie]

Differentiating smooth group homomorphisms [fcap-001Q]

Definition. The differential of a smooth homomorphism [liu2016lie, Section 2.1, p. 9] [fcap-001R]

Lemma. The differential preserves the Lie bracket [liu2016lie, Section 2.1, p. 9] [fcap-001S]

Theorem. The Lie functor [liu2016lie, Section 2.2, p. 10] [fcap-001T]

Theorem. Naturality of the exponential [liu2016lie, Section 2.4, pp. 12--13] [fcap-001U]

Proof.

Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]

Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]
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Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]

Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]

Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]

soEquivQuadratic soEquivQuadratic_lie_ι
L∃∀N

Definition. The differential of a smooth homomorphism [liu2016lie, Section 2.1, p. 9] [fcap-001R]