$\mathrm{Spin}(4)$Euclidean spin group in four dimensions

SU(2) × SU(2)

• Sporadic isogenies to orthogonal groupsSection 2.3: SU(2) × SU(2) → SO(4) on quaternionsOpen source

6

• Math 210C. Centers of simply connected semisimple groupsPage 4: D₂ = A₁ × A₁; dim Spin(4)=3+3=6Open source

ℤ₂ × ℤ₂

• Math 210C. Centers of simply connected semisimple groupsPage 4: Spin(4) ≅ SU(2) × SU(2) with center μ₂ × μ₂Open source

0

• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
• Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source

Yes

• Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
• Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

Yes

• Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
• Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

2

• Math 210C. Centers of simply connected semisimple groupsPage 4: D₂ = A₁ × A₁; rank 1+1=2Open source

Irreducibles are pairs (jL,jR), of dimension (2jL+1)(2jR+1). Descent to SO(4) requires jL+jR ∈ ℤ.

• Conrad and Landesman · Compact Lie groupsLecture 27: representations of a product groupOpen source
• Haber · Spin covers and subgroup examplesPages 7–8: SU(2) × SU(2) and the diagonal order-two SO(4) kernelOpen source

Spin(4) is diffeomorphic to S³ × S³. The kernel of the map to SO(4) is the diagonal ℤ₂.

• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
• Sporadic isogenies to orthogonal groupsSection 2.3: quaternionic product model and diagonal kernelOpen source

Euclidean Weyl spinors, self-dual two-forms, and four-dimensional geometry.

• Haber · Spin covers and subgroup examplesApplications to the indicated symmetry and representation categoryOpen source

group:su-2

• Sporadic isogenies to orthogonal groupsSection 2.3: SU(2) × SU(2) → SO(4) on quaternionsOpen source

group:sl-2-c

• Sporadic isogenies to orthogonal groupsSection 2.3: SU(2) × SU(2) → SO(4) on quaternionsOpen source

algebra:so-4-compact-r

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:so-4-compact-r

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

map:spin-4-to-so-4

• Sporadic isogenies to orthogonal groupsSection 2.3: quaternionic SO(4) covering constructionOpen source
• Haber · Spin covers and subgroup examplesPages 7–8: SU(2) × SU(2) and SO(4)Open source