Editing Simple Lie group (section)

=== Complex ===

{{See also|Complex Lie group}}
{{sort-under}}
{| class="wikitable sortable sort-under"
|-
!
! Real dimension
! {{verth|Real rank}}
! {{verth|Maximal compact<br>subgroup}}
! Fundamental<br>group
! class="unsortable" | Outer auto&shy;morphism<br>group
! class="unsortable" | Other names
! Dimension of<br>symmetric space
! class="unsortable" | Compact<br>symmetric space
! class="unsortable" | Non-Compact<br>symmetric space
|-
! ''A''<sub>''n''</sub><br/>(''n'' ≥ 1) complex
| 2''n''(''n'' + 2)
| ''n''
| ''A''<sub>''n''</sub>
| Cyclic, order {{math|''n'' + 1}}
| 2 if {{math|1=''n'' = 1}},<br/>4 (noncyclic) if {{math|''n'' ≥ 2}}.
| '''[[projective special linear group|projective complex special linear group]]'''<br>PSL<sub>''n''+1</sub>(''C'')
| ''n''(''n'' + 2)
| Compact group ''A''<sub>''n''</sub>
| Hermitian forms on ''C''<sup>''n''+1</sup>
with fixed volume.
|-
! ''B''<sub>n</sub><br/>(''n'' ≥ 2) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''B''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
| '''complex [[special orthogonal group]]'''<br>SO<sub>2''n''+1</sub>('''C''')
| ''n''(2''n'' + 1)
| Compact group ''B''<sub>n</sub>
| 
|-
! ''C''<sub>''n''</sub><br/>(''n'' ≥ 3) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''C''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
| '''projective complex [[symplectic group]]'''<br>PSp<sub>2''n''</sub>('''C''')
| ''n''(2''n'' + 1)
| Compact group ''C''<sub>n</sub>
| 
|-
! ''D''<sub>''n''</sub><br/>(''n'' ≥ 4) complex
| 2''n''(2''n'' &minus; 1)
| ''n''
| ''D''<sub>''n''</sub>
| Order 4 (cyclic when ''n'' is odd)
| Noncyclic of order 4 for {{math|''n'' > 4}}, or the product of a group of order 2 and the symmetric group ''S''<sub>3</sub> when {{math|1=''n'' = 4}}.
| '''projective complex special orthogonal group'''<br>PSO<sub>2''n''</sub>('''C''')
| ''n''(2''n'' &minus; 1)
| Compact group ''D''<sub>n</sub>
| 
|-
! ''E''<sub>6</sub> complex
| 156
| 6
| ''E''<sub>6</sub>
| 3
| Order 4 (non-cyclic)
| 
| 78
| Compact group ''E''<sub>6</sub>
| 
|-
! ''E''<sub>7</sub> complex
| 266
| 7
| ''E''<sub>7</sub>
| 2
| Order 2 (complex conjugation)
| 
| 133
| Compact group ''E''<sub>7</sub>
| 
|-
! ''E''<sub>8</sub> complex
| 496
| 8
| ''E''<sub>8</sub>
| 1
| Order 2 (complex conjugation)
| 
| 248
| Compact group ''E''<sub>8</sub>
| 
|-
! ''F''<sub>4</sub> complex
| 104
| 4
| ''F''<sub>4</sub>
| 1
| 2
| 
| 52
| Compact group ''F''<sub>4</sub>
| 
|-
! ''G''<sub>2</sub> complex
| 28
| 2
| ''G''<sub>2</sub> 
| 1
| Order 2 (complex conjugation)
| 
| 14
| Compact group ''G''<sub>2</sub>
| 
|}