Classification tables of irreducible Riemannian symmetric spaces (Cartan's list)

Intuition Beginner

Some curved spaces are more symmetric than others. The roundest of all are the symmetric spaces, where a reflection at every point is a symmetry 07.04.07. It turns out there are not infinitely many fundamentally different building blocks of this kind. Elie Cartan wrote down the complete list, and it reads like a periodic table of the most symmetric curved spaces.

The list has a clean structure. Each entry is a space of the form "big group divided by smaller subgroup", and the entries come in mirror-image pairs: a compact one (positively curved, like a sphere) and a matching open one (negatively curved, like the hyperbolic plane). The two halves of each pair share the same underlying skeleton, so once you know one you essentially know the other.

The families are labelled with two-letter codes such as AI, BDI, or CII. There are seven such families coming from the familiar matrix groups, plus a short bonus list of twelve special spaces tied to the five exceptional groups. One of those, called FII, is the Cayley projective plane, a space that only makes sense because of the octonions.

Visual Beginner

Picture a wall chart, like the periodic table, but the cells hold spaces instead of elements. The columns are the families AI, AII, AIII, BDI, DIII, CI, CII. Each cell shows the quotient written as a fraction of two groups, together with two numbers: the dimension of the space and its rank (how many independent flat directions it contains). A separate small panel on the side lists the twelve exceptional spaces.

Each cell secretly comes in two colours: a compact version and its open mirror image. The chart is the master key for looking up any single most-symmetric space and reading off its basic numbers at a glance.

Worked example Beginner

Take the real Grassmannian: the space of all $2$-dimensional planes through the origin in $5$-dimensional space. In Cartan's chart this is the family BDI, written as the quotient $SO(5)/(SO(2)\times SO(3))$.

Step 1. Read the recipe. The big group $SO(5)$ is all rotations of $5$-dimensional space. The subgroup $SO(2)\times SO(3)$ rotates the chosen plane and its complement separately. Sliding the plane around fills out the whole Grassmannian.

Step 2. Read the dimension. A $2$-plane in $5$-space is pinned down by how it tilts into the remaining $3$ directions, giving $2\times 3=6$ degrees of freedom. So the dimension is $6$.

Step 3. Read the rank. The rank counts how many planes you can tilt at once independently; here that number is $2$. This matches the entry BDI with $p=2$, $q=3$ in the table.

What this tells us: every box in the chart is a recipe (a quotient) plus a pair of numbers (dimension and rank), and the Grassmannians are the most down-to-earth members of the list.

Check your understanding Beginner

Formal definition Intermediate+

An irreducible Riemannian symmetric space is a simply-connected Riemannian symmetric space $M=G/K$ 07.04.07 whose isotropy representation of $K$ on the tangent space $T_{o}M\cong \mathfrak{p}$ is irreducible, equivalently whose orthogonal symmetric Lie algebra $(\mathfrak{g},\theta )$ has $\mathfrak{g}$ simple (in the compact and non-compact cases) or one-dimensional (the Euclidean degenerate case). Cartan's theorem organises these into four types.

Definition (the four types). Let $(\mathfrak{g},\theta )$ be an effective irreducible orthogonal symmetric Lie algebra with $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$.

• Type I (compact, $\mathfrak{g}$ compact simple): $G/K$ with $G$ compact simple, $K$ the fixed-group of an involution. Non-negative curvature.
• Type II (compact group manifold): $M=G$ itself, viewed as $(G\times G)/\Delta G$, for $G$ compact simple. Here $K=\Delta G$ is the diagonal.
• Type III (non-compact, $\mathfrak{g}$ non-compact simple): $G^{\ast }/K$ with $G^{\ast }$ a non-compact real form, $K$ maximal compact. Non-positive curvature.
• Type IV (non-compact group manifold): the complex group $G_{\mathbb{C}}$ regarded as a real symmetric space $G_{\mathbb{C}}/G$, dual to Type II.

Definition (the duality). To an orthogonal symmetric Lie algebra $(\mathfrak{g},\theta )=(\mathfrak{k}\oplus \mathfrak{p},\theta )$ of compact type associate its dual ${\mathfrak{g}}^{\ast }=\mathfrak{k}\oplus i\mathfrak{p}$ inside the complexification ${\mathfrak{g}}_{\mathbb{C}}$, with the same involution restricted. This exchanges Type I with Type III and Type II with Type IV, fixing $\mathfrak{k}$ and reversing the sign of the Killing form on the tangent space, hence the curvature sign. The correspondence $G/K\leftrightarrow G^{\ast }/K$ is a bijection on the level of the tables.

Each space carries a rank (dimension of a maximal flat totally geodesic submanifold, equal to $\dim$ of a maximal abelian subalgebra $\mathfrak{a}\subset \mathfrak{p}$) and a restricted root system 07.04.08, the set of eigenvalues of $\mathrm{ad}(\mathfrak{a})$ acting on $\mathfrak{g}$, which is one of the types $A_r,B_r,C_r,D_r,B{C}_r,E_r,F_4,G_2$.

Counterexamples to common slips

• A space and its dual are not the same space. $SU(n)/SO(n)$ (Type I, compact) and $SL(n,\mathbb{R})/SO(n)$ (Type III, non-compact) share the isotropy $SO(n)$, the dimension, and the rank, but they are different manifolds with opposite curvature. The table lists them as one row read two ways.

• Type AIII is one row even though it contains the Hermitian Grassmannians. The Hermitian sublist 07.04.14 is a proper subset of Cartan's full list: only AIII, BDI ($p=2$), DIII, CI, EIII, EVII are Hermitian. The remaining rows (AI, AII, BDI generic, CII, and most exceptionals) carry no invariant complex structure.

• The "exceptional" spaces are not exceptions to the method. EI through G are produced by exactly the same machinery (involutions of a simple algebra); they are called exceptional only because $\mathfrak{g}$ is one of $E_6,E_7,E_8,F_4,G_2$, the five Lie algebras outside the classical series.

Key theorem with proof Intermediate+

Theorem (Cartan's classification, classical families). The irreducible Riemannian symmetric spaces of compact type whose orthogonal symmetric Lie algebra has $\mathfrak{g}$ a classical simple algebra are exhausted by the following seven families. Each row also names its non-compact dual, real dimension, rank, and restricted-root type.

Cartan label	Compact $U/K$	Non-compact dual $G^{\ast }/K$	${\dim }_{\mathbb{R}}$	rank	restricted roots
AI	$SU(n)/SO(n)$	$SL(n,\mathbb{R})/SO(n)$	$(n-1)(n+2)/2$	$n-1$	$A_{n-1}$
AII	$SU(2n)/Sp(n)$	$S{U}^{\ast }(2n)/Sp(n)$	$(n-1)(2n+1)$	$n-1$	$A_{n-1}$
AIII	$S(U(p)\times U(q))\backslash SU(p+q)$	$SU(p,q)/S(U(p)\times U(q))$	$2pq$	$\min (p,q)$	$B{C}_{\min }$ or $C_{\min }$
BDI	$SO(p+q)/(SO(p)\times SO(q))$	$S{O}_0(p,q)/(SO(p)\times SO(q))$	$pq$	$\min (p,q)$	$B_{\min }$ or $D_{\min }$
DIII	$SO(2n)/U(n)$	$S{O}^{\ast }(2n)/U(n)$	$n(n-1)$	$[n/2]$	$C_{[n/2]}$ or $BC$
CI	$Sp(n)/U(n)$	$Sp(n,\mathbb{R})/U(n)$	$n(n+1)$	$n$	$C_n$
CII	$Sp(p+q)/(Sp(p)\times Sp(q))$	$Sp(p,q)/(Sp(p)\times Sp(q))$	$4pq$	$\min (p,q)$	$B{C}_{\min }$ or $C_{\min }$

In addition, every compact simple Lie group $G$ is itself an irreducible symmetric space of type II, $G=(G\times G)/\Delta G$, with non-compact dual the complex group $G_{\mathbb{C}}/G$ of type IV.

Proof. By the correspondence between simply-connected symmetric spaces and effective orthogonal symmetric Lie algebras 07.04.07, it suffices to classify the involutive automorphisms $\theta$ of a classical simple Lie algebra $\mathfrak{g}$ up to conjugacy, then read off $\mathfrak{k}={\mathfrak{g}}^{\theta }$ and $\mathfrak{p}$ the $(-1)$-eigenspace.

Step 1: reduce to real forms. An involution $\theta$ of the compact form $\mathfrak{u}$ is the same datum as a real form ${\mathfrak{g}}^{\ast }=\mathfrak{k}\oplus i\mathfrak{p}$ of the complexification ${\mathfrak{u}}_{\mathbb{C}}$, via the construction of the dual. Hence the compact-type spaces are in bijection with the non-compact real forms of ${\mathfrak{u}}_{\mathbb{C}}$, which are classified by 07.04.05 through Satake diagrams. This is the engine: the symmetric-space tables are a relabelling of the real-form tables.

Step 2: enumerate for each classical type. For ${\mathfrak{g}}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ (type $A_{n-1}$) the involutions fall into three conjugacy classes: $X\mapsto \bar{X}$ (fixed algebra $\mathfrak{so}(n)$, giving AI), $X\mapsto J\bar{X}{J}^{-1}$ with $J$ the symplectic form (fixed algebra $\mathfrak{sp}(n)$, giving AII), and $X\mapsto -I_{p,q}{X}^{\ast }{I}_{p,q}$ (fixed algebra $\mathfrak{s}(\mathfrak{u}(p)\oplus \mathfrak{u}(q))$, giving AIII). For $\mathfrak{so}(n,\mathbb{C})$ the involutions split the orthogonal space into a $p,q$ block decomposition (BDI) or introduce a complex structure (DIII). For $\mathfrak{sp}(n,\mathbb{C})$ they give CI (complex structure on the symplectic space) and CII (quaternionic block decomposition).

Step 3: compute the invariants. For each fixed pair $(\mathfrak{g},\mathfrak{k})$ the dimension is $\dim \mathfrak{g}-\dim \mathfrak{k}$, the rank is the dimension of a maximal abelian $\mathfrak{a}\subset \mathfrak{p}$, and the restricted roots are the nonzero weights of $\mathfrak{a}$ on $\mathfrak{g}$ 07.04.08. Substituting the matrix realisations gives the entries tabulated above. Type II arises from the involution swapping the two factors of $\mathfrak{g}\oplus \mathfrak{g}$, whose fixed algebra is the diagonal. $\square$

Bridge. This classification builds toward the exceptional half of the list in the Master tier and appears again in the harmonic analysis on $G/K$, where the restricted-root column dictates the form of the spherical functions. The foundational reason the table is finite is that the real-form classification 07.04.05 is finite, and this is exactly the relabelling carried out in Step 1; the compact-noncompact duality is dual to the operation $\mathfrak{p}\mapsto i\mathfrak{p}$ on the orthogonal symmetric Lie algebra. Putting these together, the central insight is that one diagram-level enumeration controls both the curved geometry and its mirror image, so reading a single row two ways generalises the sphere-hyperboloid duality to every entry on the chart.

Exercises Intermediate+

Advanced results Master

Theorem 1 (the twelve exceptional spaces). The irreducible Riemannian symmetric spaces of compact type whose simple algebra $\mathfrak{g}$ is exceptional are exactly twelve, organised by the underlying algebra:

Label	Compact $U/K$	${\dim }_{\mathbb{R}}$	rank	roots
EI	$E_6/Sp(4)$	$42$	$6$	$E_6$
EII	$E_6/(SU(6)\cdot SU(2))$	$40$	$4$	$F_4$
EIII	$E_6/(Spin(10)\cdot U(1))$	$32$	$2$	$B{C}_2$
EIV	$E_6/F_4$	$26$	$2$	$A_2$
EV	$E_7/SU(8)$	$70$	$7$	$E_7$
EVI	$E_7/(Spin(12)\cdot SU(2))$	$64$	$4$	$F_4$
EVII	$E_7/(E_6\cdot U(1))$	$54$	$3$	$C_3$
EVIII	$E_8/Spin(16)$	$128$	$8$	$E_8$
EIX	$E_8/(E_7\cdot SU(2))$	$112$	$4$	$F_4$
FI	$F_4/(Sp(3)\cdot SU(2))$	$28$	$4$	$F_4$
FII	$F_4/Spin(9)$	$16$	$1$	$B{C}_1$
G	$G_2/SO(4)$	$8$	$2$	$G_2$

Each row has a non-compact dual obtained by passing to the matching real form (for instance EIV's dual is $E_{6(-26)}/F_4$, EIII's dual is the Hermitian domain $E_{6(-14)}/(Spin(10)\cdot U(1))$). EIII and EVII are the only Hermitian members; G is the $G_2$ space $G_2/SO(4)$ of quaternion-Kahler type; FII is the Cayley projective plane.

Theorem 2 (completeness of the list). Every irreducible simply-connected Riemannian symmetric space is isometric, up to scaling, to exactly one entry in the classical table, the type-II/IV group-manifold rows, or the exceptional table, together with its dual. There are no others. The Euclidean space ${\mathbb{R}}^n$ is the degenerate reducible case excluded by irreducibility.

Theorem 3 (duality is a curvature-reversing isometry of skeletons). For each compact entry $U/K$ with curvature operator $R$, the non-compact dual $G^{\ast }/K$ has curvature operator $-R$ on the identified tangent space $\mathfrak{p}$. Consequently $U/K$ and $G^{\ast }/K$ have identical isotropy representations, identical ranks, and identical restricted-root combinatorics, differing only in the global sign of curvature and in topology (compact versus contractible).

Theorem 4 (rank-one spaces). The irreducible symmetric spaces of rank one are precisely the projective spaces and their hyperbolic duals: $S^n$ and ${\mathbb{H}}^n$ (BDI with $\min (p,q)=1$), ${\mathbb{CP}}^n$ and ${\mathbb{CH}}^n$ (AIII, $\min =1$), ${\mathbb{HP}}^n$ and ${\mathbb{HH}}^n$ (CII, $\min =1$), and the two-point Cayley pair ${\mathbb{OP}}^2=$ FII and its dual. These are exactly the two-point-homogeneous spaces.

Synthesis. Cartan's list is the foundational reason that the most symmetric geometries form a finite, fully tabulated catalogue rather than an unmanageable continuum. The central insight is that the geometric classification is a relabelling of the algebraic classification of real forms 07.04.05, and this is exactly why each row carries a clean dimension-rank-root signature. Putting these together with the duality of Theorem 3, the compact and non-compact halves are dual to one another at the level of the orthogonal symmetric Lie algebra, and the Hermitian sublist 07.04.14 is the proper subset detectable by a one-dimensional centre. The bridge is the restricted-root column 07.04.08: it generalises the rank-one projective-space pattern of Theorem 4 to every entry, so that reading the table is reading the harmonic analysis. This pattern recurs throughout the master list, where the exceptional rows obey the same diagram-driven bookkeeping as the classical ones.

Full proof set Master

Proposition 1 (dimension and rank of EVII). The exceptional Hermitian space EVII $=E_7/(E_6\cdot U(1))$ has real dimension $54$ and rank $3$, and is Hermitian with restricted root system $C_3$.

Proof. The dimension is $\dim E_7-\dim (E_6\cdot U(1))=133-(78+1)=54$. For the Hermitian property, the isotropy algebra is ${\mathfrak{e}}_6\oplus \mathfrak{u}(1)$, whose centre is the $\mathfrak{u}(1)$ summand, one-dimensional and non-zero; by the centre criterion 07.04.14 the space admits an invariant complex structure, so it is Hermitian. For the rank, the tangent space $\mathfrak{p}$ is the $56$-dimensional minuscule representation of ${\mathfrak{e}}_6$ tensored against the complex structure, and a maximal abelian $\mathfrak{a}\subset \mathfrak{p}$ has dimension $3$; equivalently EVII is a tube-type domain whose Jordan algebra is the $27$-dimensional exceptional Jordan algebra, of rank $3$. The restricted roots are then of type $C_3$ (tube type forces $C_r$ rather than $B{C}_r$). $\square$

Proposition 2 (G $=G_2/SO(4)$ has rank two and root system $G_2$). The space $G_2/SO(4)$ has real dimension $8$, rank $2$, and restricted root system of type $G_2$.

Proof. The dimension is $\dim G_2-\dim SO(4)=14-6=8$. The involution $\theta$ defining this space has fixed algebra $\mathfrak{so}(4)=\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, and the $(-1)$-eigenspace $\mathfrak{p}$ is the $8$-dimensional $(2,4)$ representation of $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$. A maximal abelian subalgebra $\mathfrak{a}\subset \mathfrak{p}$ has dimension $2$, since $G_2$ has split rank $2$ and this space is the split real form's symmetric space. The eigenvalues of $\mathrm{ad}(\mathfrak{a})$ on ${\mathfrak{g}}_2$ reproduce the full $G_2$ root system: six short and six long roots, with the characteristic $30^{\circ }$ angle. The centre of $\mathfrak{so}(4)$ is zero, so this space is not Hermitian; it is instead quaternion-Kahler, the smallest of the Wolf spaces. $\square$

Connections Master

• Classification of real forms 07.04.05. The engine behind every table in this unit. An involution of a compact simple algebra is the same datum as a non-compact real form of its complexification, so the symmetric-space rows are a faithful relabelling of the Satake-diagram classification. Each Cartan label AI through CII and EI through G is read directly off a real form, and the dimension-rank-root data are computed from the corresponding maximal compactly embedded subalgebra.

• Riemannian symmetric space 07.04.07. Supplies the correspondence between simply-connected symmetric spaces and effective orthogonal symmetric Lie algebras that turns geometry into algebra. The four types (compact I, group II, non-compact III, complex IV) and the duality $G/K\leftrightarrow G^{\ast }/K$ are stated there in general and instantiated here as concrete tables; the curvature-sign reversal of Theorem 3 is the geometric face of the operation $\mathfrak{p}\mapsto i\mathfrak{p}$.

• Hermitian symmetric space 07.04.14. The Hermitian sublist is the proper subset of Cartan's list detected by a one-dimensional centre of $\mathfrak{k}$: AIII, BDI with $p=2$, DIII, CI, EIII, EVII. This unit places that sublist inside the full classification, showing it is the complex-analytic stratum of an otherwise real-geometric catalogue, and the tube-versus-non-tube distinction matches the $C_r$-versus-$B{C}_r$ entries here.

• Restricted root system 07.04.08. Provides the final column of every table. The restricted roots are the eigenvalues of a maximal flat acting on the algebra, and their type ($A_r,B_r,C_r,D_r,B{C}_r,E_r,F_4,G_2$) controls the spherical functions and the structure of invariant differential operators on each space, making the classification tables the index for harmonic analysis on $G/K$.

Historical & philosophical context Master

Elie Cartan produced the complete classification in his two-part memoir of 1926-1927 (Bull. Soc. Math. France 54 and 55) ^[Cartan1926], building directly on his own earlier classification of the real simple Lie algebras. The achievement was conceptual as much as computational: Cartan recognised that the condition of parallel curvature forces a symmetric space to be a quotient $G/K$ by an involution, so that the entire problem of classifying these geometries collapses onto the algebraic problem of classifying involutions of simple Lie algebras. The resulting tables, with their seven classical families and twelve exceptional spaces, were among the first places where the exceptional groups $E_6,E_7,E_8,F_4,G_2$ appeared as the symmetry groups of concrete geometric objects rather than as abstract curiosities.

The modern presentation, with the dimension-rank-restricted-root tables in the form used here, is due to Helgason, whose monograph ^{[Helgason1978]} tabulates all four types together with their duals and remains the standard reference. Berger's 1957 thesis ^[Berger1957] reorganised the non-compact side and clarified the isotropy-irreducible cases, while Loos ^[Loos1969] gave a self-contained derivation of the compact tables. The philosophical lesson recurs across mathematics: a rigidity condition (here $\nabla R=0$) can be so strong that the objects satisfying it form a finite, completely enumerable list, turning an open-ended geometric question into a closed combinatorial one.

Bibliography Master

@article{Cartan1926,
  author = {Cartan, Elie},
  title = {Sur une classe remarquable d'espaces de {R}iemann},
  journal = {Bull. Soc. Math. France},
  volume = {54--55},
  year = {1926},
  pages = {214--264 and 114--134},
}

@book{Helgason1978,
  author = {Helgason, Sigurdur},
  title = {Differential Geometry, Lie Groups, and Symmetric Spaces},
  publisher = {Academic Press},
  year = {1978},
}

@article{Berger1957,
  author = {Berger, Marcel},
  title = {Les espaces sym\'etriques noncompacts},
  journal = {Ann. Sci. \'Ecole Norm. Sup.},
  volume = {74},
  year = {1957},
  pages = {85--177},
}

@book{Loos1969,
  author = {Loos, Ottmar},
  title = {Symmetric Spaces, Vol. II: Compact Spaces and Classification},
  publisher = {W. A. Benjamin},
  year = {1969},
}