Real forms of a complex semisimple Lie algebra

Intuition Beginner

A complex semisimple Lie algebra is a symmetry algebra built over the complex numbers. A real form is a way of cutting that complex object down to a real one of half the dimension, so that allowing complex coefficients again rebuilds exactly the original. One complex algebra can be cut in several genuinely different ways. These are its real forms.

Think of the complex algebra as a single ideal shape, and each real form as one physical silhouette of that shape. The silhouettes can look very different from each other --- some are compact and bounded, some stretch off to infinity --- yet they all cast the same complex shadow.

A familiar small case lives behind the rotation group. The compact algebra of rotations of three-dimensional space and the algebra of three-dimensional space-time boosts both complexify to the same complex algebra. They are two real forms of one complex symmetry.

Visual Beginner

Picture one large complex Lie algebra at the centre, drawn as a sphere. Several real forms branch off from it, each drawn as a flat disc slicing through the sphere at a different angle. Every disc has half the real dimension of the sphere, and pushing any disc back out into the complex directions refills the whole sphere.

Two of the discs are marked specially. One is the compact real form, the most rounded slice, where the bounded directions fill everything. The opposite extreme is the split form, the most stretched slice, with the largest possible unbounded part. The remaining discs are the intermediate forms, each carrying its own signature of bounded against unbounded directions.

Worked example Beginner

Take the complex algebra of two-by-two complex matrices with zero trace. Its real dimension is six. We exhibit two real forms, each of real dimension three.

Step 1. The first real form is the algebra of two-by-two real matrices with zero trace. Allowing complex entries recovers the full complex algebra, so this is a real form. It is the split form, and it is unbounded: it contains diagonal matrices with large real diagonal entries.

Step 2. The second real form is the algebra of two-by-two skew-hermitian matrices with zero trace. These are the infinitesimal rotations of complex two-space. This form is compact and bounded: it is the rotation algebra of three-dimensional real space in disguise.

Step 3. Both forms have real dimension three, and both rebuild the same six-dimensional complex algebra when complex coefficients are restored.

What this tells us: a single complex algebra carries several real shapes. Here one shape is the bounded rotation form and the other is the unbounded split form, and they are the two ends of a range of signatures.

Check your understanding Beginner

Formal definition Intermediate+

Let ${\mathfrak{g}}_{\mathbb{C}}$ be a complex semisimple Lie algebra.

Definition (Real form). A real form of ${\mathfrak{g}}_{\mathbb{C}}$ is a real Lie subalgebra ${\mathfrak{g}}_0\subset {\mathfrak{g}}_{\mathbb{C}}$ such that the natural map ${\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0\to {\mathfrak{g}}_{\mathbb{C}}$ is an isomorphism of real Lie algebras; equivalently $\mathbb{C}{\otimes }_{\mathbb{R}}{\mathfrak{g}}_0\cong {\mathfrak{g}}_{\mathbb{C}}$.

Definition (Conjugation). Each real form ${\mathfrak{g}}_0$ determines a conjugation $\sigma :{\mathfrak{g}}_{\mathbb{C}}\to {\mathfrak{g}}_{\mathbb{C}}$, the conjugate-linear involutive automorphism with $\sigma (X+iY)=X-iY$ for $X,Y\in {\mathfrak{g}}_0$. Then ${\mathfrak{g}}_0=\{Z\in {\mathfrak{g}}_{\mathbb{C}}:\sigma (Z)=Z\}$ is its fixed-point set. Conversely every conjugate-linear involutive automorphism arises from a unique real form. So real forms correspond to conjugations.

Compact and split forms. Among the real forms two are distinguished. The compact real form $\mathfrak{u}$ (Weyl) has negative-definite Killing form; its conjugation is denoted $\tau$. The split real form (or normal real form) ${\mathfrak{g}}_{\mathrm{spl}}$ is the one whose maximal abelian subalgebra of the noncompact part $\mathfrak{p}$ has the largest possible dimension, equal to the rank of ${\mathfrak{g}}_{\mathbb{C}}$. It is spanned over $\mathbb{R}$ by a Chevalley basis 07.06.18. The forms strictly between these two are the intermediate forms.

Cartan involution attached to a real form. Each real form ${\mathfrak{g}}_0$ carries a Cartan involution $\theta$ 07.04.03, an ordinary (complex-linear sense on ${\mathfrak{g}}_0$, real-linear) involutive automorphism with $B_{\theta }(X,Y)=-\kappa (X,\theta Y)$ positive definite, giving ${\mathfrak{g}}_0=\mathfrak{k}\oplus \mathfrak{p}$. One may always arrange $\theta =\sigma \tau {|}_{{\mathfrak{g}}_0}$, so $\sigma$ and $\tau$ commute and $\theta$ is the restriction of $\sigma \tau$. The pair $(\sigma ,\tau )$ of commuting conjugations encodes the form.

Counterexamples to common slips

• Not every real subalgebra of half the dimension is a real form. The real span of a chosen real basis is a real form only when the structure constants in that basis are real and the span is bracket-closed; a generic half-dimensional real subspace fails to be a subalgebra.

• The compact form is not the only form with a compact piece. Every real form ${\mathfrak{g}}_0=\mathfrak{k}\oplus \mathfrak{p}$ has a maximal compact subalgebra $\mathfrak{k}$. What singles out the compact real form is that $\mathfrak{p}=0$, so the entire algebra is compact, not merely a subalgebra.

• Distinct conjugations can give isomorphic real forms. Two conjugations ${\sigma }_1,{\sigma }_2$ give isomorphic real forms exactly when they are conjugate under $\mathrm{Aut}({\mathfrak{g}}_{\mathbb{C}})$. Counting real forms up to isomorphism means counting these conjugacy classes, not the conjugations themselves.

Key theorem with proof Intermediate+

Theorem (Classification of real forms by Cartan involutions). Let ${\mathfrak{g}}_{\mathbb{C}}$ be a complex semisimple Lie algebra with compact real form $\mathfrak{u}$ and conjugation $\tau$. There are bijections among the following three sets:

$$\{\text{real forms of }{\mathfrak{g}}_{\mathbb{C}}\}/\cong ⟷\{\text{Cartan involutions of }\mathfrak{u}\}/\mathrm{Int}(\mathfrak{u})⟷\{\text{involutive automorphisms of }\mathfrak{u}\}/\mathrm{Int}(\mathfrak{u}).$$

A real form ${\mathfrak{g}}_0$ with conjugation $\sigma$ (chosen to commute with $\tau$) maps to the involution $\theta =\sigma \tau$ of $\mathfrak{u}$.

Proof.

Step 1: from a real form to an involution of $\mathfrak{u}$. Let ${\mathfrak{g}}_0$ be a real form with conjugation $\sigma$. By the existence theorem for Cartan involutions 07.04.03, after applying an inner automorphism we may assume $\sigma (\mathfrak{u})=\mathfrak{u}$, equivalently $\sigma \tau =\tau \sigma$. Then $\theta :=\sigma \tau$ is a complex-linear automorphism (product of two conjugate-linear maps), and ${\theta }^2=\sigma \tau \sigma \tau ={\sigma }^2{\tau }^2=\mathrm{id}$ using commutativity. Since $\sigma$ and $\tau$ both preserve $\mathfrak{u}$, the restriction $\theta {|}_{\mathfrak{u}}$ is an involutive automorphism of $\mathfrak{u}$.

Step 2: from an involution of $\mathfrak{u}$ to a real form. Let $\theta$ be an involutive automorphism of $\mathfrak{u}$, with eigenspace splitting $\mathfrak{u}=\mathfrak{k}\oplus i{\mathfrak{p}}_0$ into the $+1$ and $-1$ eigenspaces (writing the $-1$ eigenspace as $i{\mathfrak{p}}_0$ with ${\mathfrak{p}}_0$ real). Extend $\theta$ complex-linearly to ${\mathfrak{g}}_{\mathbb{C}}$. Set ${\mathfrak{g}}_0=\mathfrak{k}\oplus {\mathfrak{p}}_0$ where ${\mathfrak{p}}_0=i\cdot (-1\text{-eigenspace})$. Then ${\mathfrak{g}}_0$ is a real subalgebra: the bracket relations $[\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k}$, $[\mathfrak{k},{\mathfrak{p}}_0]\subset {\mathfrak{p}}_0$, $[{\mathfrak{p}}_0,{\mathfrak{p}}_0]\subset \mathfrak{k}$ follow because $\theta$ is an automorphism and the factor of $i^2=-1$ from ${\mathfrak{p}}_0=i(\cdot )$ flips the closure of the noncompact bracket into $\mathfrak{k}$. The Killing form is negative definite on $\mathfrak{k}\subset \mathfrak{u}$ and positive definite on ${\mathfrak{p}}_0$ (it picks up a sign from the factor $i$), so ${\mathfrak{g}}_0$ has a Cartan involution and is a real form with ${\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0=\mathfrak{u}\oplus i\mathfrak{u}={\mathfrak{g}}_{\mathbb{C}}$.

Step 3: the maps are mutually inverse on classes. Starting from ${\mathfrak{g}}_0$, building $\theta =\sigma \tau$, then forming $\mathfrak{k}\oplus i{\mathfrak{p}}_0⇝\mathfrak{k}\oplus {\mathfrak{p}}_0$ returns ${\mathfrak{g}}_0$, since $\mathfrak{k}$ is the $\sigma$-fixed part of $\mathfrak{u}$ and ${\mathfrak{p}}_0$ is the $\sigma$-fixed part of $i\mathfrak{u}$. Starting from $\theta$, the real form ${\mathfrak{g}}_0$ has conjugation $\sigma$ with $\sigma \tau =\theta$ by construction. The ambiguity in Step 1 (choosing the inner automorphism that makes $\sigma \tau =\tau \sigma$) is exactly the conjugacy of Cartan involutions 07.04.03, so the maps descend to bijections on $\mathrm{Int}(\mathfrak{u})$-conjugacy classes. The third set equals the second because the Cartan condition is automatic for involutions of a compact algebra: every involutive automorphism of $\mathfrak{u}$ is a Cartan involution there.

$\square$

Bridge. This classification builds toward the symmetric-space tables of 07.04.13, where each real form indexes one noncompact symmetric space and its compact dual. The foundational reason the count is finite is that involutions of $\mathfrak{u}$ up to conjugacy are governed by finite combinatorial data on the Dynkin diagram, and this is exactly the bridge between the analytic problem of cutting a complex algebra into real shapes and the discrete problem of marking a diagram. The central insight is that the compact form $\mathfrak{u}$ serves as a universal anchor: every other real form appears again in the list as a single involution of $\mathfrak{u}$, and the construction generalises the single Cartan involution of 07.04.03 into a full enumeration. Putting these together, the geometry of 07.04.07 and the algebra of 07.04.01 meet here.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Vogan diagrams). Fix the compact form $\mathfrak{u}$ and a maximal torus. To each real form attach a Vogan diagram: the Dynkin diagram of ${\mathfrak{g}}_{\mathbb{C}}$, together with (i) the involution of the diagram induced by $\theta$ on a $\theta$-stable maximally compact Cartan subalgebra, drawn as arrows joining swapped nodes, and (ii) a painting of those simple roots fixed by the involution that are noncompact imaginary. Every real form arises from a Vogan diagram, and two diagrams give isomorphic forms exactly when related by the equivalence generated by diagram automorphisms and the Borel-de Siebenthal painting moves at a single node. This makes the classification combinatorial and recovers Cartan's original labels such as $A\mathrm{I},A\mathrm{II},A\mathrm{III}$.

Theorem 2 (Satake diagrams, alternative encoding). The Satake diagram of a real form records the same data through restricted roots 07.04.03: simple roots that vanish on the maximal split abelian $\mathfrak{a}$ are painted black, the remaining white nodes carry the restricted-root structure, and arrows join white nodes identified by the conjugation $-\sigma$ on the root system. Satake and Vogan diagrams encode equivalent information; Satake is adapted to the noncompact split data and the restricted root system, Vogan to the compact maximal torus and the painted imaginary roots.

Theorem 3 (Classification tables). The real forms of each complex simple type are:

• $A_{n-1}=\mathfrak{sl}(n,\mathbb{C})$: compact $\mathfrak{su}(n)$; split $\mathfrak{sl}(n,\mathbb{R})$ ($A\mathrm{I}$); $\mathfrak{su}(p,q)$, $p+q=n$ ($A\mathrm{III}$); and for $n=2m$ even, $\mathfrak{sl}(m,\mathbb{H})={\mathfrak{su}}^{\ast }(2m)$ ($A\mathrm{II}$).
• $B_n=\mathfrak{so}(2n+1,\mathbb{C})$: compact $\mathfrak{so}(2n+1)$; the forms $\mathfrak{so}(p,q)$ with $p+q=2n+1$, the split form being $\mathfrak{so}(n,n+1)$.
• $C_n=\mathfrak{sp}(n,\mathbb{C})$: compact $\mathfrak{sp}(n)$; split $\mathfrak{sp}(n,\mathbb{R})$; and $\mathfrak{sp}(p,q)$, $p+q=n$.
• $D_n=\mathfrak{so}(2n,\mathbb{C})$: compact $\mathfrak{so}(2n)$; $\mathfrak{so}(p,q)$, $p+q=2n$, split $\mathfrak{so}(n,n)$; and ${\mathfrak{so}}^{\ast }(2n)$.
• $G_2$: compact $G_{2(-14)}$; split $G_{2(2)}$.
• $F_4$: compact; split $F_{4(4)}$; and $F_{4(-20)}$.
• $E_6$: compact; split $E_{6(6)}$; $E_{6(2)}$; $E_{6(-14)}$; $E_{6(-26)}$.
• $E_7$: compact; split $E_{7(7)}$; $E_{7(-5)}$; $E_{7(-25)}$.
• $E_8$: compact; split $E_{8(8)}$; $E_{8(-24)}$.

The subscript is the signature $\dim \mathfrak{p}-\dim \mathfrak{k}$ of the Killing form, an isomorphism invariant.

Theorem 4 (Two extremes are unique). For each complex simple ${\mathfrak{g}}_{\mathbb{C}}$ the compact real form and the split real form are each unique up to isomorphism. The compact form is the empty Vogan diagram (no painted nodes, identity involution where the diagram has no automorphism); the split form is the one whose restricted root system has the same rank as the absolute system, with Satake diagram all white and no arrows beyond the diagram automorphism.

Theorem 5 (Cartan's count). The number of real forms of each complex simple algebra is finite and matches Cartan's 1914 list: types $A_{n-1}$ give $\lfloor n/2\rfloor +2$ forms (for $n\ge 3$), the exceptional $G_2,F_4,E_6,E_7,E_8$ give $2,3,5,4,3$ forms respectively, including the compact one in each count.

Synthesis. The classification of real forms is the foundational reason that the apparently analytic problem of slicing a complex symmetry into real shapes becomes a finite combinatorial exercise. The central insight is that fixing the compact form $\mathfrak{u}$ converts every real form into a single involution of one fixed algebra, and this is exactly the move that lets the painted Dynkin diagram carry all the information. Putting these together, the Vogan and Satake encodings are dual bookkeeping of one structure: Vogan generalises the compact-torus viewpoint, Satake the split viewpoint, and the bridge between them is the conjugation acting on roots. The pattern recurs in 07.04.13, where each real form indexes a symmetric space, so the count of forms is the count of irreducible noncompact symmetric spaces; the split-to-compact range appears again as the curvature range from negative through flat to positive, and the whole table generalises the single Cartan involution of 07.04.03 into the full atlas.

Full proof set Master

Proposition 1 (Uniqueness of the compact real form). Any two compact real forms of a complex semisimple Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$ are conjugate under $\mathrm{Int}({\mathfrak{g}}_{\mathbb{C}})$, and in particular isomorphic.

Proof. Let ${\mathfrak{u}}_1,{\mathfrak{u}}_2$ be compact real forms with conjugations ${\tau }_1,{\tau }_2$. Each ${\tau }_i$ is a conjugate-linear involutive automorphism, and $-\kappa (X,{\tau }_{i}Y)$ is a positive-definite Hermitian form on ${\mathfrak{g}}_{\mathbb{C}}$. The composite $\rho ={\tau }_1{\tau }_2$ is a complex-linear automorphism, self-adjoint and positive definite with respect to the inner product $-\kappa (\cdot ,{\tau }_1\cdot )$.

Form the positive square root $\psi ={\rho }^{1/2}$, an automorphism of ${\mathfrak{g}}_{\mathbb{C}}$ in the connected automorphism group, hence in $\mathrm{Int}({\mathfrak{g}}_{\mathbb{C}})$. A computation with the spectral decomposition of $\rho$ shows $\psi {\tau }_2{\psi }^{-1}={\tau }_1$: indeed $\rho {\tau }_2={\tau }_1{\tau }_2{\tau }_2={\tau }_1$ and ${\tau }_2\rho ={\tau }_2{\tau }_1{\tau }_2$, and conjugate-linearity gives ${\tau }_2\rho {\tau }_2={\rho }^{-1}$, so ${\tau }_2$ inverts $\rho$ and therefore commutes with $\psi$ up to the inversion that yields $\psi {\tau }_2{\psi }^{-1}={\rho }^{1/2}{\tau }_2{\rho }^{-1/2}={\rho }^{1/2}{\rho }^{1/2}{\tau }_2=\rho {\tau }_2={\tau }_1$. Hence $\psi ({\mathfrak{u}}_2)={\mathfrak{u}}_1$, so the two compact forms are conjugate under an inner automorphism. $\square$

Proposition 2 (Real forms biject with conjugacy classes of involutions of $\mathfrak{u}$). Isomorphism classes of real forms of ${\mathfrak{g}}_{\mathbb{C}}$ are in bijection with conjugacy classes of involutive automorphisms of the compact form $\mathfrak{u}$ under $\mathrm{Int}(\mathfrak{u})$.

Proof. Fix $\mathfrak{u}$ with conjugation $\tau$. Given a real form ${\mathfrak{g}}_0$ with conjugation $\sigma$, Proposition 1 lets us replace $\sigma$ by an inner-conjugate so that $\sigma \tau =\tau \sigma$; then $\theta =\sigma \tau$ restricts to an involution of $\mathfrak{u}$. If two choices of the adjusting inner automorphism are made, the resulting involutions differ by an element of $\mathrm{Int}(\mathfrak{u})$ that commutes with $\tau$, so the $\mathrm{Int}(\mathfrak{u})$-class of $\theta$ is well-defined.

Conversely, an involution $\theta$ of $\mathfrak{u}$ extends complex-linearly and the form $\sigma =\theta \tau$ is conjugate-linear and involutive (as $\theta$ and $\tau$ commute), so its fixed set is a real form ${\mathfrak{g}}_0$. The two assignments invert each other on classes: the freedom in choosing $\sigma$ within its isomorphism class corresponds precisely to $\mathrm{Int}(\mathfrak{u})$-conjugation of $\theta$ by the conjugacy of Cartan involutions 07.04.03. Hence the assignment ${\mathfrak{g}}_0\mapsto [\theta ]$ is a bijection on classes. $\square$

Connections Master

• Cartan-Weyl classification 07.04.01. The classification of complex semisimple Lie algebras by Dynkin diagrams is the input to the real classification. Each complex type supports a fixed family of real forms, so the real list is a refinement: one complex diagram fans out into several painted diagrams, one per real form, and the absolute root system of 07.04.01 is the backbone on which Vogan and Satake paintings are placed.

• Cartan involution 07.04.03. A real form is detected by a single Cartan involution, and the present unit enumerates exactly which involutions can occur. The existence-and-conjugacy theorem of 07.04.03 supplies the well-definedness of the bijection between forms and involutions of the compact algebra, so 07.04.03 constructs one form while this unit counts them all.

• Riemannian symmetric space 07.04.07. Each noncompact real form ${\mathfrak{g}}_0=\mathfrak{k}\oplus \mathfrak{p}$ gives a symmetric space $G/K$ whose curvature sign is read off from the Killing form on $\mathfrak{p}$. The table of real forms is therefore the table of irreducible symmetric spaces, with the compact-split range matching the positive-to-negative curvature range, linking this algebraic enumeration directly to the geometry of 07.04.07.

• Root space decomposition 07.06.18. The split real form is spanned over $\mathbb{R}$ by a Chevalley basis drawn from the root space decomposition, and the restricted roots that label Satake diagrams are projections of absolute roots onto the split abelian subalgebra. The decomposition of 07.06.18 is thus the coordinate system in which every real form is written.

Historical & philosophical context Master

Elie Cartan classified the real forms of the complex simple Lie algebras in his 1914 memoir (Ann. Sci. Ecole Norm. Sup. 31) ^[Cartan1914], completing the real counterpart of his earlier complex classification and producing the labels $A\mathrm{I}$ through $E\mathrm{VIII}$ still in use. His method tracked the signature of the Killing form, and the list of forms became the list of irreducible Riemannian symmetric spaces once the geometric correspondence was understood. Hermann Weyl's construction of the compact real form (the unitary trick) supplied the universal anchor that later treatments use to convert real forms into involutions of one fixed compact algebra.

The combinatorial encodings came later. Satake (and Araki's refinement, J. Math. Osaka City Univ. 13, 1962) ^[Araki1962] introduced the restricted-root diagrams that bear Satake's name, while David Vogan's painted Dynkin diagrams gave the compact-torus encoding now standard in textbook accounts. Helgason's 1978 monograph and Knapp's Lie Groups Beyond an Introduction synthesised both viewpoints, and Onishchik-Vinberg ^{[OnishchikVinberg1990]} gave the algebraic-group perspective. Philosophically the subject exemplifies a recurring theme: an infinite-looking analytic classification (real structures on a complex object) collapses to a finite decorated-diagram problem, the same compression that organises root systems and reflection groups.

Bibliography Master

@article{Cartan1914,
  author = {Cartan, Elie},
  title = {Les groupes reels simples finis et continus},
  journal = {Ann. Sci. Ecole Norm. Sup.},
  volume = {31},
  year = {1914},
  pages = {263--355},
}

@article{Araki1962,
  author = {Araki, Shoro},
  title = {On root systems and an infinitesimal classification of irreducible symmetric spaces},
  journal = {J. Math. Osaka City Univ.},
  volume = {13},
  year = {1962},
  pages = {1--34},
}

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

@book{Knapp2002,
  author = {Knapp, Anthony W.},
  title = {Lie Groups Beyond an Introduction},
  publisher = {Birkhauser},
  year = {2002},
  edition = {2nd},
}

@book{OnishchikVinberg1990,
  author = {Onishchik, Arkady L. and Vinberg, Ernest B.},
  title = {Lie Groups and Algebraic Groups},
  publisher = {Springer},
  year = {1990},
}