Remark. Algebraic extension versus topological cover [fcap-000Z]
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The extensions in Theorem [fcap-001D] and Theorem [fcap-001F] are exact sequences of abstract groups. They record a surjective homomorphism and its two-element kernel. No topology on the groups is used.

For real quadratic spaces of arbitrary signature, Gallier proves that the two maps \[\operatorname {Pin}(p,q)\longrightarrow O(p,q),\qquad \operatorname {Spin}(p,q)\longrightarrow SO(p,q)\] are topological double covers [gallier2014clifford, Section 1.8, Proposition 1.21, pp. 39--40].

Gallier then treats the compact groups: \(\operatorname {Spin}(n)\) is path-connected for \(n\geq 2\), and for \(n\geq 3\) it is simply connected and hence the universal cover of \(SO(n)\) [gallier2014clifford, Section 1.8, pp. 40--42]. These claims are not being extended here to arbitrary signature. Kostant identifies the differential of the complex Spin cover with the quadratic Clifford Lie algebra [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]. Those conclusions require topological or Lie-theoretic structure not supplied by an abstract group extension. They are not asserted by the four cards above.

Remark. Algebraic extension versus topological cover [fcap-000Z]
✍️source

Reference. Clifford algebras, clifford groups, and a generalization of the quaternions [gallier2014clifford]

Reference. Clifford algebra analogue of the Hopf--Koszul--Samelson theorem [kostant1997clifford]

From reflection generation to Pin and Spin [fcap-000U]

Lemma. Fixed-subspace correction by reflections [cartan1981theory, Section 10, pp. 10--12] [fcap-000V]

exists_mem_subgroup_mul_eqOn_sup_span_singleton_of_reflection_mem
L∃∀N

Proof.

Lemma. Pin-range refinement of the correction [cartan1981theory, Section 10, pp. 10--12]; [lawson2016spin, I.2, (2.26) and Theorem 2.9, pp. 18--19] [fcap-0018]

exists_mem_range_pinToOrthogonal_mul_eqOn_sup_span_singleton
L∃∀N

Theorem. Cartan--Dieudonne generation [lawson2016spin, I.2, Theorem 2.7, p. 17] [fcap-000W]

subgroup_eq_top_of_reflection_mem
L∃∀N

Proof.

Theorem. Surjectivity of the Pin action [lawson2016spin, I.2, Theorem 2.9, pp. 18--19] [fcap-000X]

pinToOrthogonal_surjective_of_isSquare pinToOrthogonal_surjective
L∃∀N

Example. A planar rotation as two reflections [lawson2016spin, I.2, (2.24)--(2.26), p. 18] [fcap-000Y]

Lemma. Determinant parity detects the even lift [lawson2016spin, I.2, Theorems 2.7 and 2.9, pp. 17--19] [fcap-001A]

det_reflection mem_even_of_det_pinToOrthogonal_eq_one
L∃∀N

Proof.

Theorem. Surjectivity of the Spin action [lawson2016spin, I.2, Theorem 2.9, pp. 18--19] [fcap-001B]

spinToSpecialOrthogonal_surjective_of_isSquare spinToSpecialOrthogonal_surjective
L∃∀N

Lemma. The real Spin kernel is the scalar pair [gallier2014clifford, Section 1.8, Proposition 1.21, pp. 39--40] [fcap-001C]

zmodTwoMulEquivKerSpinToSpecialOrthogonal
L∃∀N

Theorem. The algebraic Spin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001D]

spinDoubleCoverOfSurjective spinDoubleCover
L∃∀N

Lemma. The real Pin kernel is the scalar pair [gallier2014clifford, Section 1.8, Proposition 1.21, pp. 39--40] [fcap-001E]

zmodTwoMulEquivKerPinToOrthogonal
L∃∀N

Theorem. The algebraic Pin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001F]

pinDoubleCoverOfSurjective pinDoubleCover
L∃∀N

Remark. Algebraic extension versus topological cover [fcap-000Z]
✍️source

Theorem. The algebraic Pin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001F]

pinDoubleCoverOfSurjective pinDoubleCover
L∃∀N

Theorem. The algebraic Spin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001D]

spinDoubleCoverOfSurjective spinDoubleCover
L∃∀N