$\mathrm{SO}_0(2,4)$Identity component of the orthogonal group of signature (2,4)

The identity component of the determinant-one real 6 × 6 matrices preserving a quadratic form with signature (2,4).

• 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

15

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

2

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

C₂

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

Z × C₂

• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
• Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
• Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

No

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

Yes

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

No

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

Yes

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

SU(2,2)/{±I}

• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open 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
• Sporadic isogenies to orthogonal groupsSection 2.11: SU(2,2) double coverOpen source

su(2,2) ≅ so(2,4)

• 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

SO(2) × SO(4)

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
• Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source

A Cartan decomposition gives maximal compact subgroup SO(2) × SO(4), hence π₁ = Z × C₂. The matrix −I lies in the identity component because both signature blocks have even dimension; the center is {±I}. Its spin double cover SU(2,2) still has π₁ = Z.

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open 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
• Sporadic isogenies to orthogonal groupsSection 2.11: SU(2,2) double coverOpen source

The vector 6 is an ordinary representation of this matrix group. The chiral spinor 4 of SU(2,2) does not descend through the order-two covering kernel. For a spacetime application, specify separately whether the action uses this group, its projective quotient, or a further covering group.

• Sporadic isogenies to orthogonal groupsSection 2.11: exterior-square orthogonal action and its kernelOpen source

algebra:su-2-2-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:su-2-2-r

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

map:su-2-2-to-so-2-4

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open 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
• Sporadic isogenies to orthogonal groupsSection 2.11: SU(2,2) double coverOpen source

group:su-2-2

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open 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
• Sporadic isogenies to orthogonal groupsSection 2.11: SU(2,2) double coverOpen source