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    "slug": "so-4",
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    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [
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    ],
    "tags": [
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      "Reductive algebra",
      "Global form"
    ],
    "summary": "(SU(2) × SU(2))/diagonal C₂. The real orthogonal 4 × 4 matrices of determinant 1.",
    "definition": "The real orthogonal 4 × 4 matrices of determinant 1.",
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      },
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            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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            "source": "etingof-coxeter",
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            "source": "conrad",
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        "key": "center",
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
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        "key": "simple-algebra",
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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            "source": "garrett-isogenies",
            "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)"
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        "key": "lie-algebra",
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      {
        "id": "topology",
        "title": "Global structure and topology",
        "paragraphs": [
          {
            "text": "The universal cover Spin(4) = SU(2) × SU(2) acts on quaternions by x ↦ axb⁻¹. Its kernel is {(1,1),(−1,−1)}. The center of SO(4) is {±I}.",
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                "source": "conrad",
                "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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              {
                "source": "garrett-isogenies",
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            "text": "An irreducible Spin(4) representation (jL,jR) descends to SO(4) exactly when jL + jR is an integer. The dimension is (2jL + 1)(2jR + 1).",
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                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
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                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              },
              {
                "source": "haber",
                "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)"
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            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
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            "source": "etingof-real-forms",
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        "type": "has-map",
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            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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          {
            "source": "garrett-isogenies",
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            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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            "source": "garrett-isogenies",
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          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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          "source": "conrad",
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          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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      "assumptions": [],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:so-4#abelian",
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      "key": "abelian",
      "label": "Abelian",
      "value": "No",
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          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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      "revision": "2026.11-workbench.4"
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      "id": "group:so-4#simple-algebra",
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      "key": "simple-algebra",
      "label": "Simple real Lie algebra",
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          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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          "source": "conrad",
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          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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          "source": "garrett-isogenies",
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      "revision": "2026.11-workbench.4"
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      "id": "group:so-4#lie-algebra",
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          "source": "etingof-lie-notes",
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          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
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          "source": "haber",
          "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)"
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          "source": "etingof-lie-notes",
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          "source": "etingof-real-forms",
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          "source": "conrad-centers",
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          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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        {
          "source": "garrett-isogenies",
          "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)"
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      "label": "Specified connected covering group",
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          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        },
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)"
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  "sources": [
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      "id": "haber",
      "title": "Haber · Spin covers and subgroup examples",
      "authors": [
        "Howard E. Haber"
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      "locatorHint": "Spring 2017; Spin(4) on pages 7–8, SU(2) and SO(3) subgroups of SU(3) on pages 14–15.",
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    },
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      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
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        "Aaron Landesman"
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    },
    {
      "id": "conrad-centers",
      "title": "Math 210C. Centers of simply connected semisimple groups",
      "authors": [
        "Brian Conrad"
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