Editing Simple Lie group (section)

==Simple Lie groups of small dimension==

The following table lists some Lie groups with simple Lie algebras of small 
dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.

{| class="wikitable sortable"
|-
!Dim
!Groups
!
!Symmetric space
!Compact dual
!Rank
!Dim
|-
|1
|{{mathbb|R}}, ''S''<sup>1</sup> = U(1) = SO<sub>2</sub>({{mathbb|R}}) = Spin(2)
|Abelian
|Real line
|
|0
|1
|-
|3
| ''S''<sup>3</sup> = Sp(1) = SU(2)=Spin(3), SO<sub>3</sub>({{mathbb|R}}) = PSU(2) 
|Compact
|
|
|
|
|-
|3
|SL<sub>2</sub>({{mathbb|R}}) = Sp<sub>2</sub>({{mathbb|R}}), SO<sub>2,1</sub>({{mathbb|R}})
|Split, Hermitian, hyperbolic
|Hyperbolic plane <math>\mathbb{H}^2</math>
|Sphere ''S''<sup>2</sup>
|1
|2
|-
|6
|SL<sub>2</sub>({{mathbb|C}}) = Sp<sub>2</sub>({{mathbb|C}}), SO<sub>3,1</sub>({{mathbb|R}}), SO<sub>3</sub>({{mathbb|C}})
|Complex
|Hyperbolic space <math>\mathbb{H}^3</math>
|Sphere ''S''<sup>3</sup>
|1
|3
|-
|8
|SL<sub>3</sub>({{mathbb|R}})
|Split
|Euclidean structures on <math>\mathbb{R}^3</math>
|Real structures on <math>\mathbb{C}^3</math>
|2
|5
|-
|8
|SU(3)
|Compact
|
|
|
|
|-
|8
|SU(1,2)
|Hermitian, quasi-split, quaternionic
|Complex hyperbolic plane
|Complex projective plane
|1
|4
|-
|10
|Sp(2) = Spin(5), SO<sub>5</sub>({{mathbb|R}})
|Compact
|
|
|
|
|-
|10
|SO<sub>4,1</sub>({{mathbb|R}}), Sp<sub>2,2</sub>({{mathbb|R}})
|Hyperbolic, quaternionic
|Hyperbolic space <math>\mathbb{H}^4</math>
|Sphere ''S''<sup>4</sup>
|1
|4
|-
|10
|SO<sub>3,2</sub>({{mathbb|R}}), Sp<sub>4</sub>({{mathbb|R}})
|Split, Hermitian
|Siegel upper half space
|Complex structures on <math>\mathbb{H}^2</math>
|2
|6
|-
|14
|''G''<sub>2</sub> 
|Compact
|
|
|
|
|-
|14
|''G''<sub>2</sub> 
|Split, quaternionic
|Non-division quaternionic subalgebras of non-division octonions
|Quaternionic subalgebras of octonions
|2
|8
|-
|15
|SU(4) = Spin(6), SO<sub>6</sub>({{mathbb|R}})
|Compact
|
|
|-
|15
|SL<sub>4</sub>({{mathbb|R}}), SO<sub>3,3</sub>({{mathbb|R}})
|Split
|{{mathbb|R}}<sup>3</sup> in {{mathbb|R}}<sup>3,3</sup>
|Grassmannian ''G''(3,3)
|3
|9
|-
|15
|SU(3,1), SO*(6)
|Hermitian
|Complex hyperbolic space
|Complex projective space
|1
|6
|-
|15
|SU(2,2), SO<sub>4,2</sub>({{mathbb|R}})
|Hermitian, quasi-split, quaternionic
|{{mathbb|R}}<sup>2</sup> in {{mathbb|R}}<sup>2,4</sup>
|Grassmannian ''G''(2,4)
|2
|8
|-
|15
|SL<sub>2</sub>({{mathbb|H}}), SO<sub>5,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^5</math>
|Sphere ''S''<sup>5</sup>
|1
|5
|-
|16
|SL<sub>3</sub>({{mathbb|C}})
|Complex
|
|SU(3)
|2
|8
|-
|20
|SO<sub>5</sub>({{mathbb|C}}), Sp<sub>4</sub>({{mathbb|C}})
|Complex
|
|Spin<sub>5</sub>({{mathbb|R}})
|2
|10
|-
|21
|SO<sub>7</sub>({{mathbb|R}})
|Compact
|
|
|-
|21
|SO<sub>6,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^6</math>
|Sphere ''S''<sup>6</sup>
|-
|21
|SO<sub>5,2</sub>({{mathbb|R}})
|Hermitian
|
|
|-
|21
|SO<sub>4,3</sub>({{mathbb|R}})
|Split, quaternionic
|
|
|-
|21
|Sp(3)
|Compact
|
|
|-
|21
|Sp<sub>6</sub>({{mathbb|R}})
|Split, hermitian
|
|
|-
|21
|Sp<sub>4,2</sub>({{mathbb|R}})
|Quaternionic
|
|
|-
|24
|SU(5)
|Compact
|
|
|-
|24
|SL<sub>5</sub>({{mathbb|R}})
|Split
|
|
|-
|24
|SU<sub>4,1</sub>
|Hermitian
|
|
|-
|24
|SU<sub>3,2</sub>
|Hermitian, quaternionic
|
|
|-
|28
|SO<sub>8</sub>({{mathbb|R}})
|Compact
|
|
|-
|28
|SO<sub>7,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^7</math>
|Sphere ''S''<sup>7</sup>
|-
|28
|SO<sub>6,2</sub>({{mathbb|R}}), SO<sup>∗</sup><sub>8</sub>({{mathbb|R}})
|Hermitian
|
|
|-
|28
|SO<sub>5,3</sub>({{mathbb|R}})
|Quasi-split
|
|
|-
|28
|SO<sub>4,4</sub>({{mathbb|R}})
|Split, quaternionic
|
|
|-
|28
|''G''<sub>2</sub>({{mathbb|C}})
|Complex
|
|
|-
|30
|SL<sub>4</sub>({{mathbb|C}})
|Complex
|
|
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