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{{short description|Wikipedia list article}}
{{mergeto|Simple Lie group|date=December 2019}}
{{Lie groups |Simple}}

In [[Mathematics|mathematics]], the [[Simple Lie group|simple Lie group]]s were first classified by Wilhelm Killing and later perfected by [[Biography:Élie Cartan|Élie Cartan]]. This classification is often referred to as Killing-Cartan classification. 

The list of simple Lie groups can be used to read off the list of [[Simple Lie algebra|simple Lie algebra]]s and Riemannian symmetric spaces.  See also the [[Table of Lie groups|table of Lie groups]] for a smaller list of groups that commonly occur in [[Physics:Theoretical physics|theoretical physics]], and the [[Astronomy:Bianchi classification|Bianchi classification]] for groups of dimension at most 3. 

== Simple Lie groups ==

Unfortunately, there is no universally accepted definition of a [[Simple Lie group|simple Lie group]]. In particular, it is not always defined as a Lie group that is [[Simple group|simple]] as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether '''R''' is a simple Lie group. 

The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed ''connected'' normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but '''R''' is not simple. 

In this article the connected simple Lie groups with trivial center are listed.  Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a [[Covering space#universal covers|universal cover]], whose center is the [[Fundamental group|fundamental group]] of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.

== Simple Lie algebras ==
{{main|Simple Lie algebra}}
The [[Lie group–Lie algebra correspondence|Lie algebra of a simple Lie group]] is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with [[Triviality (mathematics)|trivial]] center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)

Over the complex numbers the semisimple Lie algebras are classified by their [[Dynkin diagram]]s, of types "ABCDEFG". If ''L'' is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless ''L'' is already the complexification of a Lie algebra, in which case the complexification of ''L'' is a product of two copies of ''L''. This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra.

== Symmetric spaces ==

Symmetric spaces are classified as follows. 

First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)

Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces). 

The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to  each ''non-compact'' simple Lie group ''G'',
one compact and one non-compact. The non-compact one is a cover of the quotient of ''G'' by a maximal compact subgroup ''H'', and the compact one is a cover of the quotient of
the compact form of ''G'' by the same subgroup ''H''. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.

== Hermitian symmetric spaces ==

A symmetric space with a compatible complex structure is called Hermitian.  The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces.

The four families are the types A&nbsp;III, B&nbsp;I and D&nbsp;I for {{nowrap|1=''p'' = 2}}, D&nbsp;III, and C&nbsp;I, and the two exceptional ones are types E&nbsp;III and E&nbsp;VII of complex dimensions 16 and 27.

== Notation ==

<math> \mathbb {R, C, H, O} </math>&nbsp; stand for the real numbers, complex numbers, quaternions, and [[Octonion|octonion]]s.

In the symbols such as ''E''<sub>6</sub><sup>&minus;26</sup> for the exceptional groups, the exponent &minus;26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup.

The fundamental group listed in the table below is the fundamental group of the simple group with trivial center. 
Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group).

== List ==

=== Abelian ===


{| class="wikitable"
|-
! width=100|
! Dimension
! Outer automorphism group
! Dimension of symmetric space
! Symmetric space
! Remarks
|-class="sorttop"
! R (Abelian)
| 1
| '''R'''<sup>∗</sup>
| 1
| '''R'''
| {{Ref|Note†|†}}
|}

=== Compact ===


{| 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> ({{nowrap|''n'' ≥ 1}}) compact
| ''n''(''n'' + 2)
| 0
| Cyclic, order {{nowrap|''n'' + 1}}
| 1 if {{nowrap|1=''n'' = 1}}, 2 if {{nowrap|''n'' > 1}}.
| '''projective special unitary group'''<br>{{nowrap|PSU(''n'' + 1)}}
| ''A''<sub>1</sub> is the same as ''B''<sub>1</sub> and ''C''<sub>1</sub>
|-
! ''B''<sub>''n''</sub> ({{nowrap|''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> ({{nowrap|''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> ({{nowrap|''n'' ≥ 4}}) compact
| ''n''(2''n'' &minus; 1)
| 0
| Order 4 (cyclic when ''n'' is odd).
| 2 if {{nowrap|''n'' > 4}}, ''S''<sub>3</sub> if {{nowrap|1=''n'' = 4}}
|style="white-space:nowrap"| '''projective special [[Orthogonal group|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.
|}

=== Split ===


{| class="wikitable sortable"
|-
! width=100|
! 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>''n''</sub> I (''n'' ≥ 1) split
| ''n''(''n'' + 2)
| ''n''
| ''D''<sub>''n''/2</sub> or ''B''<sub>(''n''&minus;1)/2</sub>
|style="white-space:nowrap"| Infinite cyclic if ''n'' = 1<br>2 if ''n'' ≥ 2
| 1 if ''n'' = 1<br>2 if ''n'' ≥ 2.
|style="white-space:nowrap"| '''projective [[Special linear group|special linear group]]'''<br>PSL<sub>''n''+1</sub>(R)
| ''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 {{nowrap|1=''n'' = 1}}, in which case it is the 2-sphere. 
| Euclidean structures on ''R''<sup>''n''+1</sup>. Hermitian if {{nowrap|1=''n'' = 1}}, when it is the upper half plane or unit complex disc.
| 
|-
! ''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)
| ''n''(''n'' + 1)
|
|
| ''B''<sub>1</sub> is the same as ''A''<sub>1</sub>.
|-
! ''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|symplectic group]]'''<br>PSp<sub>2''n''</sub>(''R''), PSp(2''n'',''R''), PSp(2''n''), PSp(''n'',''R''), PSp(''n'')
| ''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>.
|-
! ''D''<sub>''n''</sub> I (''n'' ≥ 4) split
| ''n''(2''n'' - 1)
| ''n''
| SO(''n'')SO(''n'')
| Order 4 if ''n'' odd, 8 if ''n'' even
| 2 if {{nowrap|''n'' > 4}}, ''S''<sub>3</sub> if {{nowrap|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.
|-
! ''E''<sub>6</sub><sup>6</sup> I split
| 78
| 6
| ''C''<sub>4</sub>
| Order 2
| Order 2
| E I
| 42 
| 
| 
| 
|-
! ''E''<sub>7</sub><sup>7</sup> V split
| 133
| 7
| ''A''<sub>7</sub>
| Cyclic, order 4
| Order 2
| 
| 70
| 
| 
| 
|-
! ''E''<sub>8</sub><sup>8</sup> VIII split
| 248
| 8
| ''D''<sub>8</sub>
| 2
| 1
| E VIII
| 128
| 
| 
| @ [[E8 (mathematics)|E8]]
|-
! ''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.
| 
|-
! ''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.
| 
|}

=== Complex ===


{| class="wikitable sortable"
|-
! width=100|
! Real 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
|-
! ''A''<sub>''n''</sub> (''n'' ≥ 1) complex
| 2''n''(''n'' + 2)
| ''n''
| ''A''<sub>''n''</sub>
| Cyclic, order {{nowrap|''n'' + 1}}
| 2 if {{nowrap|1=''n'' = 1}}, 4 (noncyclic) if {{nowrap|''n'' ≥ 2}}.
|style="white-space:nowrap"| '''projective complex [[Special linear group|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> (''n'' ≥ 2) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''B''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
|style="white-space:nowrap"| '''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> (''n'' ≥ 3) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''C''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
|style="white-space:nowrap"| '''projective complex [[Symplectic group|symplectic group]]'''<br>PSp<sub>2''n''</sub>('''C''')
| ''n''(2''n'' + 1)
| Compact group ''C''<sub>n</sub>
| 
|-
! ''D''<sub>''n''</sub> (''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 {{nowrap|''n'' > 4}}, or the product of a group of order 2 and the symmetric group ''S''<sub>3</sub> when {{nowrap|1=''n'' = 4}}.
|style="white-space:nowrap"| '''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>
| 
|}

=== Others ===

{| class="wikitable sortable"
|-
! width=100|
! 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''&minus;1</sub> II<br>(''n'' ≥ 2)
| (2''n'' &minus; 1)(2''n'' + 1)
| ''n'' &minus; 1
| ''C''<sub>''n''</sub>
| Order 2
| 
| SL<sub>''n''</sub>(''H''), SU<sup>∗</sup>(2''n'')
| {{hs|zzzzzz <!-- when sorted, comes after numbers -->}}(''n'' &minus; 1)(2''n'' + 1)
| Quaternionic structures on ''C''<sup>2''n''</sup> compatible with the Hermitian structure
| Copies of quaternionic hyperbolic space (of dimension {{nowrap|''n'' &minus; 1}}) in [[Complex hyperbolic space|complex hyperbolic space]] (of dimension {{nowrap|2''n'' &minus; 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''&minus;1</sub>''A''<sub>''q''&minus;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'' &minus; 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 {{nowrap|1=''p'' = ''q''}}, split<br>If {{abs|''p''−''q''}} ≤ 2, quasi-split
|-
! ''D''<sub>''n''</sub> III<br>(''n'' ≥ 4)
| ''n''(2''n'' &minus; 1)
| ⌊''n''/2⌋
| ''A''<sub>''n''&minus;1</sub>''R''<sup>1</sup>
| Infinite cyclic
| Order 2
| ''SO''<sup>*</sup>(2n)
| ''n''(''n'' &minus; 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>&minus;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>&minus;26</sup> IV
| 78
| 2
| ''F''<sub>4</sub>
| Trivial
| Order 2
| E IV
| 26
| Set of Cayley projective planes 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>&minus;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>&minus;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>&minus;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>&minus;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.
|
|}

==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"
|-
!Dim
!Groups
!
!Symmetric space
!Compact dual
!Rank
!Dim
|-
|1
|''R'', ''S''<sup>1</sup>=U(1)=SO<sub>2</sub>(''R'')=Spin(2)
|Abelian
|Real line
|
|0
|1
|-
|3
| ''S''<sup>3</sup>=Sp(1)=SU(2)=Spin(3), SO<sub>3</sub>(''R'')=PSU(2) 
|Compact
|
|
|
|
|-
|3
|SL<sub>2</sub>(''R'')=Sp<sub>2</sub>(''R''), SO<sub>2,1</sub>(''R'')
|Split, Hermitian, hyperbolic
|Hyperbolic plane ''H''<sup>2</sup>
|Sphere ''S''<sup>2</sup>
|1
|2
|-
|6
|SL<sub>2</sub>(''C'')=Sp<sub>2</sub>(''C''), SO<sub>3,1</sub>(''R''), SO<sub>3</sub>(''C'')
|Complex
|Hyperbolic space ''H''<sup>3</sup>
|Sphere ''S''<sup>3</sup>
|1
|3
|-
|8
|SL<sub>3</sub>(''R'')
|Split
|Euclidean structures on '''R'''<sup>3</sup>
|Real structures on '''C'''<sup>3</sup>
|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>(''R'')
|Compact
|
|
|
|
|-
|10
|SO<sub>4,1</sub>(''R''), Sp<sub>2,2</sub>(''R'')
|Hyperbolic, quaternionic
|Hyperbolic space ''H''<sup>4</sup>
|Sphere ''S''<sup>4</sup>
|1
|4
|-
|10
|SO<sub>3,2</sub>(''R''),Sp<sub>4</sub>(''R'')
|Split, Hermitian
|Siegel upper half space
|Complex structures on '''H'''<sup>2</sup>
|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>(''R'')
|Compact
|
|
|-
|15
|SL<sub>4</sub>(''R''), SO<sub>3,3</sub>(''R'')
|Split
|'''R'''<sup>3</sup> in '''R'''<sup>3,3</sup>
|Grassmannian ''G''(3,3)
|3
|9
|-
|15
|SU(3,1)
|Hermitian
|Complex hyperbolic space
|Complex projective space
|1
|6
|-
|15
|SU(2,2), SO<sub>4,2</sub>(''R'')
|Hermitian, quasi-split, quaternionic
|'''R'''<sup>2</sup> in '''R'''<sup>2,4</sup>
|Grassmannian ''G''(2,4)
|2
|8
|-
|15
|SL<sub>2</sub>(H), SO<sub>5,1</sub>(''R'')
|Hyperbolic
|Hyperbolic space ''H''<sup>5</sup>
|Sphere ''S''<sup>5</sup>
|1
|5
|-
|16
|SL<sub>3</sub>(''C'')
|Complex
|
|SU(3)
|2
|8
|-
|20
|SO<sub>5</sub>(''C''), Sp<sub>4</sub>(''C'')
|Complex
|
|Spin<sub>5</sub>('''R''')
|2
|10
|-
|21
|SO<sub>7</sub>(''R'')
|Compact
|
|
|-
|21
|SO<sub>6,1</sub>(''R'')
|Hyperbolic
|Hyperbolic space ''H''<sup>6</sup>
|Sphere ''S''<sup>6</sup>
|-
|21
|SO<sub>5,2</sub>(''R'')
|Hermitian
|
|
|-
|21
|SO<sub>4,3</sub>(''R'')
|Split, quaternionic
|
|
|-
|21
|Sp(3)
|Compact
|
|
|-
|21
|Sp<sub>6</sub>(''R'')
|Split, hermitian
|
|
|-
|21
|Sp<sub>4,2</sub>(''R'')
|Quaternionic
|
|
|-
|24
|SU(5)
|Compact
|
|
|-
|24
|SL<sub>5</sub>(''R'')
|Split
|
|
|-
|24
|SU<sub>4,1</sub>
|Hermitian
|
|
|-
|24
|SU<sub>3,2</sub>
|Hermitian, quaternionic
|
|
|-
|28
|SO<sub>8</sub>(''R'')
|Compact
|
|
|-
|28
|SO<sub>7,1</sub>(''R'')
|Hyperbolic
|Hyperbolic space ''H''<sup>7</sup>
|Sphere ''S''<sup>7</sup>
|-
|28
|SO<sub>6,2</sub>(''R'')
|Hermitian
|
|
|-
|28
|SO<sub>5,3</sub>(''R'')
|Quasi-split
|
|
|-
|28
|SO<sub>4,4</sub>(''R'')
|Split, quaternionic
|
|
|-
|28
|SO<sup>∗</sup><sub>8</sub>(''R'')
|Hermitian
|
|
|-
|28
|''G''<sub>2</sub>(''C'')
|Complex
|
|
|-
|30
|SL<sub>4</sub>(''C'')
|Complex
|
|
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==Notes==
:{{Note|Note†|†}} The group '''R''' is not simple as an abstract group, and according to most (but not all) definitions this is not a simple Lie group. Most authors do not count its Lie algebra as a simple Lie algebra. It is listed here so that the list of irreducible simply connected symmetric spaces is complete. Note that '''R''' is the only such non-compact symmetric space without a compact dual (although it has a compact quotient ''S''<sup>1</sup>).

==Further reading==
* Besse, ''Einstein manifolds'' {{isbn|0-387-15279-2}}
* Helgason, ''Differential geometry, Lie groups, and symmetric spaces''. {{isbn|0-8218-2848-7}}
* Fuchs and Schweigert, ''Symmetries, Lie algebras, and representations: a graduate course for physicists.'' Cambridge University Press, 2003.  {{isbn|0-521-54119-0}}

[[Category:Lie groups]]
[[Category:Mathematics-related lists|Lie groups]]

{{Sourceattribution|List of simple Lie groups|1}}