lecture-notes
Lecture 15: Classification of Real Forms of Semisimple Lie Algebras
Pavel Etingof
Use in this reference: Linked MIT OpenCourseWare notes; definitions and classification facts are summarized in original prose.
Relevant material: Section 41, printed pages 218–225.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SU(2) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| U(1) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(4) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2,ℂ) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2,ℂ) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Aᵣ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| Aᵣ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| Aᵣ | Classical real forms | Sections 41.2–41.4: compact, split, and classical real forms |
| Bᵣ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| Bᵣ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| Bᵣ | Classical real forms | Sections 41.2–41.4: compact, split, and classical real forms |
| Cᵣ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| Cᵣ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| Cᵣ | Classical real forms | Sections 41.2–41.4: compact, split, and classical real forms |
| Dᵣ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| Dᵣ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| Dᵣ | Classical real forms | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ | Current real-form coverage | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ | Current real-form coverage | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ | Current real-form coverage | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ | Current real-form coverage | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ | Algebra, real form, global group | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ | Current real-form coverage | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(4, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(5, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| so(5, C) ≅ sp(4, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(6, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆(C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇(C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈(C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄(C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂(C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) ⊕ sl(2, C) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| C (zero bracket) | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) ⊕ C | Using the algebra record | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| u(1) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| u(2) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Simple over R | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Compact real algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Curated integrating group | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(3) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(4) ≅ so(6) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(5) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(2) ≅ so(5) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(3) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(8) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| so(10) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₆ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₇ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| e₈ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| f₄ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| g₂ compact | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2) ⊕ su(2) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, R) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(3, R) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| su(2,2) ≅ so(2,4) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sp(4, R) ≅ so(2,3) | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Real dimension | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Compact real form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | One algebra, several integrations | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Actual real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Complexification | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| sl(2, C) viewed over R | Curated global form | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(4) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(4) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSU(4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(5) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(5) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(5) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(5) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| U(2) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| U(2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| U(2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(4) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(5) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(5) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(5) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(5) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(6) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(6) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(6) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(6) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(2) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(2) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(3) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(8) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(8) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(8) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(8) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| PSO(8) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSO(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSO(8) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSO(8) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(10) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(10) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(10) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Spin(10) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(10) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(10) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(10) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO(10) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ compact | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| G₂ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ compact | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| F₄ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, simply connected | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, simply connected | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, simply connected | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, simply connected | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, simply connected | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, simply connected | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, simply connected | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, simply connected | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ compact | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ compact | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₈ compact | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, adjoint | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, adjoint | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, adjoint | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₆ compact, adjoint | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, adjoint | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, adjoint | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, adjoint | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| E₇ compact, adjoint | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Global form | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Global structure and topology | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Explicit homomorphism | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Explicit homomorphism | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(2, R) | Specified connected covering group | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| PSL(2, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Universal cover of SL(2, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| Universal cover of SL(2, R) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Universal cover of SL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Universal cover of SL(2, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Universal cover of SL(2, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SL(3, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SU(2,2) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| Sp(4, R) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(1,3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,4) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Definition | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Real rank | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Simple real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Real Lie algebra | Sections 41.2–41.4: compact, split, and classical real forms |
| SO₀(2,3) | Real-form context | Sections 41.2–41.4: compact, split, and classical real forms |