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Simple Lie group
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=== Others === {{sort-under}} {| class="wikitable sortable sort-under" |- ! ! Dimension ! Real rank ! Maximal compact<br>subgroup ! Fundamental<br>group ! class="unsortable" | Outer automorphism<br>group ! class="unsortable" | Other names ! Dimension of<br>symmetric space ! class="unsortable" | Compact<br>symmetric space ! class="unsortable" | Non-Compact<br>symmetric space ! class="unsortable" | Remarks |- ! ''A''<sub>2''n''−1</sub> II<br>(''n'' ≥ 2) | {{math|(2''n'' − 1)(2''n'' + 1)}} | ''n'' − 1 | ''C''<sub>''n''</sub> | Order 2 | | SL<sub>''n''</sub>(''H''), SU<sup>∗</sup>(2''n'') | {{hs|zzzzzz <!-- when sorted, comes after numbers -->}}{{math|(''n'' − 1)(2''n'' + 1)}} | Quaternionic structures on ''C''<sup>2''n''</sup> compatible with the Hermitian structure | Copies of [[quaternionic hyperbolic space]] (of dimension {{math|''n'' − 1}}) in [[complex hyperbolic space]] (of dimension {{math|2''n'' − 1}}). | |- !style="white-space:nowrap"| ''A''<sub>''n''</sub> III<br>(''n'' ≥ 1)<br>''p'' + ''q'' = ''n'' + 1<br>(1 ≤ ''p'' ≤ ''q'') | ''n''(''n'' + 2) | ''p'' | ''A''<sub>''p''−1</sub>''A''<sub>''q''−1</sub>''S''<sup>1</sup> | | | SU(''p'',''q''), A III | 2''pq'' |style="white-space:nowrap"| [[Hermitian symmetric space|Hermitian]].<br>Grassmannian of ''p'' subspaces of ''C''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 2; [[Quaternion-Kähler symmetric space|quaternion-Kähler]] |style="white-space:nowrap"| Hermitian.<br>Grassmannian of maximal positive definite<br>subspaces of ''C''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 2, quaternion-Kähler |style="white-space:nowrap"| If ''p''=''q''=1, split<br>If {{abs|''p''−''q''}} ≤ 1, [[quasi-split]] |- ! ''B''<sub>''n''</sub> I<br>(''n'' > 1)<br>''p''+''q'' = 2''n''+1 | ''n''(2''n'' + 1) | min(''p'',''q'') | SO(''p'')SO(''q'') | | | [[indefinite orthogonal group|SO(''p'',''q'')]] | ''pq'' |style="white-space:nowrap"| Grassmannian of ''R''<sup>''p''</sup>s in ''R''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, Projective space<br>If ''p'' or ''q'' is 2; Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler |style="white-space:nowrap"| Grassmannian of positive definite ''R''<sup>''p''</sup>s in ''R''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1, Hyperbolic space<br>If ''p'' or ''q'' is 2, Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler |style="white-space:nowrap"| If {{abs|''p''−''q''}} ≤ 1, split. |- ! ''C''<sub>''n''</sub> II<br>(''n'' > 2)<br>''n'' = ''p''+''q''<br>(1 ≤ ''p'' ≤ ''q'') | ''n''(2''n'' + 1) | min(''p'',''q'') | ''C''<sub>''p''</sub>''C''<sub>''q''</sub> | Order 2 |style="white-space:nowrap"| 1 if ''p'' ≠ ''q'', 2 if ''p'' = ''q''. | Sp<sub>2''p'',2''q''</sub>(R) | 4''pq'' |style="white-space:nowrap"| Grassmannian of ''H''<sup>''p''</sup>s in ''H''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, quaternionic projective space<br>in which case it is quaternion-Kähler. |style="white-space:nowrap"| ''H''<sup>''p''</sup>s in ''H''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1, quaternionic hyperbolic space<br>in which case it is quaternion-Kähler. | |- ! ''D''<sub>''n''</sub> I<br>(''n'' ≥ 4)<br>''p''+''q'' = 2''n'' | ''n''(2''n'' − 1) | min(''p'',''q'') | SO(''p'')SO(''q'') | | If ''p'' and ''q'' ≥ 3, order 8. | SO(''p'',''q'') | ''pq'' |style="white-space:nowrap"| Grassmannian of ''R''<sup>''p''</sup>s in ''R''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, Projective space<br>If ''p'' or ''q'' is 2 ; Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler |style="white-space:nowrap"| Grassmannian of positive definite ''R''<sup>''p''</sup>s in ''R''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1, Hyperbolic Space<br>If ''p'' or ''q'' is 2, Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler |style="white-space:nowrap"| If {{math|1=''p'' = ''q''}}, split<br>If {{abs|''p''−''q''}} ≤ 2, quasi-split |- ! ''D''<sub>''n''</sub> III<br>(''n'' ≥ 4) | ''n''(2''n'' − 1) | ⌊''n''/2⌋ | ''A''<sub>''n''−1</sub>''R''<sup>1</sup> | Infinite cyclic | Order 2 | ''SO''<sup>*</sup>(2n) | ''n''(''n'' − 1) | Hermitian.<br>Complex structures on R<sup>2''n''</sup> compatible with the Euclidean structure. | Hermitian.<br>Quaternionic quadratic forms on R<sup>2''n''</sup>. | |- ! ''E''<sub>6</sub><sup>2</sup> II<br>(quasi-split) | 78 | 4 | ''A''<sub>5</sub>''A''<sub>1</sub> | Cyclic, order 6 | Order 2 | E II | 40 | Quaternion-Kähler. | Quaternion-Kähler. | Quasi-split but not split. |- ! ''E''<sub>6</sub><sup>−14</sup> III | 78 | 2 | ''D''<sub>5</sub>''S''<sup>1</sup> | Infinite cyclic | Trivial | E III | 32 | Hermitian.<br>Rosenfeld elliptic projective plane over the complexified Cayley numbers. | Hermitian.<br>Rosenfeld hyperbolic projective plane over the complexified Cayley numbers. | |- ! ''E''<sub>6</sub><sup>−26</sup> IV | 78 | 2 | ''F''<sub>4</sub> | Trivial | Order 2 | E IV | 26 | Set of [[Cayley projective plane]]s in the projective plane over the complexified Cayley numbers. | Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers. | |- ! ''E''<sub>7</sub><sup>−5</sup> VI | 133 | 4 | ''D''<sub>6</sub>''A''<sub>1</sub> | Non-cyclic, order 4 | Trivial | ''E'' VI | 64 | Quaternion-Kähler. | Quaternion-Kähler. | |- ! ''E''<sub>7</sub><sup>−25</sup> VII | 133 | 3 | ''E''<sub>6</sub>''S''<sup>1</sup> | Infinite cyclic | Order 2 | E VII | 54 | Hermitian. | Hermitian. | |- ! ''E''<sub>8</sub><sup>−24</sup> IX | 248 | 4 | ''E''<sub>7</sub> × ''A''<sub>1</sub> | Order 2 | 1 | E IX | 112 | Quaternion-Kähler. | Quaternion-Kähler. | |- ! ''F''<sub>4</sub><sup>−20</sup> II | 52 | 1 | ''B''<sub>4</sub> (Spin<sub>9</sub>('''R''')) | Order 2 | 1 | F II | 16 | Cayley projective plane. Quaternion-Kähler. | Hyperbolic Cayley projective plane. Quaternion-Kähler. | |}
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