Editing Simple Lie group (section)

=== Split ===

{{See also|Split Lie algebra}}
{{sort-under}}
{| class="wikitable sortable sort-under"
|-
! 
! Dimension
! {{verth|va=middle|Real rank}}
! Maximal compact<br>subgroup
! Fundamental<br>group
! class="unsortable" | Outer auto&shy;morphism<br>group
! class="unsortable" | Other names
! {{verth|va=middle|Dimension of<br>symmetric space}}
! class="unsortable" | Compact<br>symmetric space
! class="unsortable" | Non-Compact<br>symmetric space
! class="unsortable" | Remarks
|-
! {{verth|va=middle|''A''<sub>''n''</sub> I (''n'' ≥ 1) split}}
| ''n''(''n'' + 2)
| ''n''
| ''D''<sub>''n''/2</sub> or ''B''<sub>(''n''&minus;1)/2</sub>
| Infinite cyclic if ''n'' = 1<br>2 if ''n'' ≥ 2
| 1 if ''n'' = 1<br>2 if ''n'' ≥ 2.
| '''[[projective special linear group]]'''<br>PSL<sub>''n''+1</sub>(R)
| {{math|{{sfrac|''n''(''n'' + 3)|2}}}}
| Real structures on ''C''<sup>''n''+1</sup> or set of RP<sup>''n''</sup> in CP<sup>''n''</sup>. Hermitian if {{math|1=''n'' = 1}}, in which case it is the 2-sphere. 
| Euclidean structures on ''R''<sup>''n''+1</sup>. Hermitian if {{math|1=''n'' = 1}}, when it is the upper half plane or unit complex disc.
| 
|-
! {{verth|va=middle|''B''<sub>''n''</sub> I (''n'' ≥ 2) split}}
| ''n''(2''n'' + 1)
| ''n''
| SO(''n'')SO(''n''+1)
| Non-cyclic, order 4
| 1
| identity component of '''[[indefinite orthogonal group|special orthogonal group]]'''<br>SO(''n'',''n''+1)
| {{math|''n''(''n'' + 1)}}
|
|
| ''B''<sub>1</sub> is the same as ''A''<sub>1</sub>.
|-
! {{verth|va=middle|''C''<sub>''n''</sub> I (''n'' ≥ 3) split}}
| ''n''(2''n'' + 1)
| ''n''
| ''A''<sub>''n''&minus;1</sub>''S''<sup>1</sup>
| Infinite cyclic
| 1
| '''projective [[symplectic group]]'''<br>PSp<sub>2''n''</sub>(''R''), PSp(2''n'',''R''), PSp(2''n''), PSp(''n'',''R''), PSp(''n'')
| {{math|''n''(''n'' + 1)}}
| Hermitian. Complex structures of ''H''<sup>''n''</sup>. Copies of complex projective space in quaternionic projective space. 
| Hermitian. Complex structures on ''R''<sup>2''n''</sup> compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space.
| ''C''<sub>2</sub> is the same as ''B''<sub>2</sub>, and ''C''<sub>1</sub> is the same as ''B''<sub>1</sub> and ''A''<sub>1</sub>.
|-
! {{verth|va=middle|''D''<sub>''n''</sub> I (''n'' ≥ 4) split}}
| ''n''(2''n'' − 1)
| ''n''
| SO(''n'')SO(''n'')
| Order 4 if ''n'' odd,<br/>8 if ''n'' even
| 2 if {{math|''n'' > 4}},<br/>''S''<sub>3</sub> if {{math|1=''n'' = 4}}
| identity component of '''projective [[indefinite orthogonal group|special orthogonal group]]'''<br>PSO(''n'',''n'')
| ''n''<sup>2</sup>
|
|
| ''D''<sub>3</sub> is the same as ''A''<sub>3</sub>, ''D''<sub>2</sub> is the same as ''A''<sub>1</sub><sup>2</sup>, and ''D''<sub>1</sub> is abelian.
|-
! {{verth|va=middle|''E''<sub>6</sub><sup>6</sup> I split}}
| 78
| 6
| ''C''<sub>4</sub>
| Order 2
| Order 2
| E I
| 42 
| 
| 
| 
|-
! {{verth|va=middle|''E''<sub>7</sub><sup>7</sup> V split}}
| 133
| 7
| ''A''<sub>7</sub>
| Cyclic, order 4
| Order 2
| 
| 70
| 
| 
| 
|-
! {{verth|va=middle|''E''<sub>8</sub><sup>8</sup> VIII split}}
| 248
| 8
| ''D''<sub>8</sub>
| 2
| 1
| E VIII
| 128
| 
| 
| @ [[E8 (mathematics)|E8]]
|-
! {{verth|va=middle|''F''<sub>4</sub><sup>4</sup> I split}}
| 52
| 4
| ''C''<sub>3</sub> × ''A''<sub>1</sub>
| Order 2
| 1
| F I
| 28
| Quaternionic projective planes in Cayley projective plane.
| Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.
| 
|-
! {{verth|va=middle|''G''<sub>2</sub><sup>2</sup> I split}}
| 14
| 2
| ''A''<sub>1</sub> × ''A''<sub>1</sub>
| Order 2
| 1
| G I
| 8
| Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler.
| Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler.
| 
|}