Editing Simple Lie group (section)

=== Compact ===

{{See also|Compact group}}

{| class="wikitable sortable"
|-
! width=100|
! Dimension
! Real rank
! Fundamental<br>group
! class="unsortable" | Outer automorphism<br>group
! class="unsortable" | Other names
! class="unsortable" | Remarks
|-
! ''A''<sub>''n''</sub> ({{math|''n'' ≥ 1}}) compact
| ''n''(''n'' + 2)
| 0
| Cyclic,<br/>order {{math|''n'' + 1}}
| 1 if {{math|1=''n'' = 1}},<br/>2 if {{math|''n'' > 1}}.
| '''[[projective special unitary group]]'''<br>{{math|PSU(''n'' + 1)}}
| ''A''<sub>1</sub> is the same as ''B''<sub>1</sub> and ''C''<sub>1</sub>
|-
! ''B''<sub>''n''</sub> ({{math|''n'' ≥ 2}}) compact
| ''n''(2''n'' + 1)
| 0
| 2
| 1
| '''[[special orthogonal group]]'''<br>SO<sub>2''n''+1</sub>(''R'')
| ''B''<sub>1</sub> is the same as ''A''<sub>1</sub> and ''C''<sub>1</sub>.<br>''B''<sub>2</sub> is the same as ''C''<sub>2</sub>.
|-
! ''C''<sub>''n''</sub> ({{math|''n'' ≥ 3}}) compact
| ''n''(2''n'' + 1)
| 0
| 2
| 1
| '''projective [[compact symplectic group]]'''<br>PSp(''n''), PSp(2''n''), PUSp(''n''), PUSp(2''n'')
| Hermitian. Complex structures of ''H''<sup>''n''</sup>. Copies of complex projective space in quaternionic projective space.
|-
! ''D''<sub>''n''</sub> ({{math|''n'' ≥ 4}}) compact
| ''n''(2''n'' &minus; 1)
| 0
| Order 4 (cyclic when ''n'' is odd).
| 2 if {{math|''n'' > 4}},<br/>''S''<sub>3</sub> if {{math|1=''n'' = 4}}
|style="white-space:math"| '''projective special [[orthogonal group]]'''<br>PSO<sub>2''n''</sub>(''R'')
| ''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.
|-
! ''E''<sub>6</sub><sup>&minus;78</sup> compact
| 78
| 0
| 3
| 2
| 
| 
|-
! ''E''<sub>7</sub><sup>&minus;133</sup> compact
| 133
| 0
| 2
| 1
| 
| 
|-
! ''E''<sub>8</sub><sup>&minus;248</sup> compact
| 248
| 0
| 1
| 1
| 
| 
|-
! ''F''<sub>4</sub><sup>&minus;52</sup> compact
| 52
| 0
| 1
| 1
| 
| 
|-
! ''G''<sub>2</sub><sup>&minus;14</sup> compact
| 14
| 0
| 1
| 1
| 
| This is the automorphism group of the Cayley algebra.
|}