Lie theory · Group

$\mathrm{PSO}(8)$ Adjoint compact group of type D₄

Spin(8)/(C₂ × C₂). The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).

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Reference coverageCompact Lie groupSimple algebraGlobal form

Definition

The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).

Definition sources

Real dimension: $28$
Compact rank: $4$
Center: $1$
Fundamental group π₁: $\mathrm{C}_{2} \times \mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Real dimension	$28$	Sources
Compact rank	$4$	Sources
Center	$1$	Sources
Fundamental group π₁	$\mathrm{C}_{2} \times \mathrm{C}_{2}$	Sources
Compact	Yes	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	Spin(8)/(C₂ × C₂)	Sources
Real Lie algebra	$\mathfrak{so}(8)$	Sources

Global structure and topology

The simply connected cover is Spin(8), and the covering kernel is its full center C₂ × C₂. Thus the fundamental group has four elements and is not cyclic. [sources]

Representation category and descent

A Spin(8) irrep descends to PSO(8) precisely when its highest weight lies in the D₄ root lattice. In particular, the vector and the two half-spin 8s do not descend, whereas the adjoint 28 does. [sources]

Names and aliases

SO(8)/{±I}

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathfrak{so}(8)$
so(8) — real Lie algebra
Real Lie algebra
$\mathfrak{so}(8)$
so(8) real form
Real-form context
$\mathrm{Spin}(8)$
Simply connected compact spin group in eight dimensions
Specified connected covering group

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
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

group:pso-8#definition

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Real dimension

28

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

group:pso-8#dimension

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Compact rank

4

Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:pso-8#rank

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Center

1

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:pso-8#center

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Fundamental group π₁

C₂ × C₂

Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen 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

group:pso-8#pi1

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Compact

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

group:pso-8#compact

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Connected

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

group:pso-8#connected

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Abelian

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

group:pso-8#abelian

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Simple real Lie algebra

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

group:pso-8#simple-algebra

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Global form

Spin(8)/(C₂ × C₂)

Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:pso-8#global-form

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Real Lie algebra

so(8)

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

group:pso-8#lie-algebra

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Global structure and topology

The simply connected cover is Spin(8), and the covering kernel is its full center C₂ × C₂. Thus the fundamental group has four elements and is not cyclic.

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:pso-8#topology-1

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Representation category and descent

A Spin(8) irrep descends to PSO(8) precisely when its highest weight lies in the D₄ root lattice. In particular, the vector and the two half-spin 8s do not descend, whereas the adjoint 28 does.

Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source
Conrad and Landesman · Compact Lie groupsAppendix T.2, printed pages 261–262, especially Remark T.2.1: D₄ center, central kernels, and trialityOpen source

group:pso-8#representations-1

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Real Lie algebra

algebra:so-8-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

group:pso-8#relation-1

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Real-form context

real-form:so-8-compact-r

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

group:pso-8#relation-2

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Specified connected covering group

group:spin-8

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:pso-8#relation-3

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