{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/real-forms/so-10-compact-r/",
  "entity": {
    "id": "real-form:so-10-compact-r",
    "slug": "so-10-compact-r",
    "name": "so(10) real form",
    "symbol": "so(10)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Compact real form"
    ],
    "summary": "Compact real form of so(10, C) with its curated integrating groups.",
    "definition": "The real skew-symmetric 10 × 10 matrices with commutator bracket.",
    "definitionEvidence": [
      {
        "source": "etingof-real-forms",
        "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
      },
      {
        "source": "etingof-lie-notes",
        "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
      }
    ],
    "properties": [
      {
        "key": "dimension",
        "label": "Real dimension",
        "value": "45",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          },
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          }
        ],
        "assumptions": []
      },
      {
        "key": "complexification",
        "label": "Complexification",
        "value": "so(10, C)",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact",
        "label": "Compact real form",
        "value": "Yes",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      },
      {
        "key": "real-rank",
        "label": "Real rank",
        "value": "0",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "global-forms",
        "title": "One algebra, several integrations",
        "paragraphs": [
          {
            "text": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.",
            "evidence": [
              {
                "source": "etingof-real-forms",
                "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
              },
              {
                "source": "conrad",
                "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "realizes-algebra",
        "target": "algebra:so-10-compact-r",
        "label": "Actual real Lie algebra",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "real-form-of",
        "target": "algebra:d-5-c",
        "label": "Complexification",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "integrates-to",
        "target": "group:spin-10",
        "label": "Curated global form",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ]
      },
      {
        "type": "integrates-to",
        "target": "group:so-10",
        "label": "Curated global form",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "real-form",
    "data": {
      "complexification": "so(10, C)",
      "complexificationIds": [
        "algebra:d-5-c"
      ],
      "compact": true,
      "realRank": 0,
      "integrationIds": [
        "group:spin-10",
        "group:so-10"
      ],
      "realAlgebraId": "algebra:so-10-compact-r"
    }
  },
  "claims": [
    {
      "id": "real-form:so-10-compact-r#definition",
      "subject": "real-form:so-10-compact-r",
      "key": "definition",
      "label": "Definition",
      "value": "The real skew-symmetric 10 × 10 matrices with commutator bracket.",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#dimension",
      "subject": "real-form:so-10-compact-r",
      "key": "dimension",
      "label": "Real dimension",
      "value": "45",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#complexification",
      "subject": "real-form:so-10-compact-r",
      "key": "complexification",
      "label": "Complexification",
      "value": "so(10, C)",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#compact",
      "subject": "real-form:so-10-compact-r",
      "key": "compact",
      "label": "Compact real form",
      "value": "Yes",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#real-rank",
      "subject": "real-form:so-10-compact-r",
      "key": "real-rank",
      "label": "Real rank",
      "value": "0",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#global-forms-1",
      "subject": "real-form:so-10-compact-r",
      "key": "global-forms-1",
      "label": "One algebra, several integrations",
      "value": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#relation-1",
      "subject": "real-form:so-10-compact-r",
      "key": "relation-1",
      "label": "Actual real Lie algebra",
      "value": "algebra:so-10-compact-r",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "relationship": "realizes-algebra",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#relation-2",
      "subject": "real-form:so-10-compact-r",
      "key": "relation-2",
      "label": "Complexification",
      "value": "algebra:d-5-c",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "relationship": "real-form-of",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#relation-3",
      "subject": "real-form:so-10-compact-r",
      "key": "relation-3",
      "label": "Curated global form",
      "value": "group:spin-10",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "relationship": "integrates-to",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "real-form:so-10-compact-r#relation-4",
      "subject": "real-form:so-10-compact-r",
      "key": "relation-4",
      "label": "Curated global form",
      "value": "group:so-10",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "relationship": "integrates-to",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
      "authors": [
        "Brian Conrad",
        "Aaron Landesman"
      ],
      "url": "https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Numbered lectures and appendices; page numbers refer to the printed pagination.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf",
      "kind": "lecture-notes",
      "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.",
      "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes."
    },
    {
      "id": "etingof-real-forms",
      "title": "Lecture 15: Classification of Real Forms of Semisimple Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec15.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Section 41, printed pages 218–225.",
      "licenseNote": "Linked MIT OpenCourseWare notes; definitions and classification facts are summarized in original prose."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "7e9d48949cc228ea5aee2a2fea6df571c7fa2d4e0ef263c113c2e9190027e88f",
    "reviewers": []
  }
}
