The Cartan Classification of Simple Lie Algebras

⚡ Key Takeaways

One-line statement: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is isomorphic to exactly one member of four infinite families $A_n, B_n, C_n, D_n$ or to one of five exceptional algebras $G_2, F_4, E_6, E_7, E_8$ .

Key result in words: The structure of a simple Lie algebra is completely encoded in a finite combinatorial object — its Dynkin diagram — and the list of admissible diagrams is short and rigid.

Where it applies: Representation theory, geometry of homogeneous spaces, the standard model and grand unification, integrable systems, singularity theory, and the classification of compact simple Lie groups and symmetric spaces.

Who/when: Wilhelm Killing (1888–1890) first obtained the list; Élie Cartan (1894 thesis) corrected and rigorously proved it. Modern treatment via root systems, Weyl, van der Waerden, Witt, Dynkin (1940s).

💡 The Core Insight

A simple Lie algebra $\mathfrak{g}$ contains a maximal abelian subalgebra $\mathfrak{h}$ (the Cartan subalgebra) whose adjoint action is simultaneously diagonalizable. The nonzero eigenvalues — linear functionals $\alpha \in \mathfrak{h}^*$ called roots — form a finite, highly symmetric configuration in a Euclidean space. The integrality forced by $\mathfrak{sl}_2$ -representation theory ( $\langle \alpha, \beta^\vee\rangle \in \mathbb{Z}$ ) and the angular rigidity it implies reduce the entire problem to a finite combinatorial enumeration of allowed diagrams. The Lie algebra is reconstructed from the diagram via Serre's relations.

DefinitionSimple and Semisimple Lie Algebras

A Lie algebra $\mathfrak{g}$ over a field $k$ is a $k$ -vector space with a bilinear bracket $[\cdot,\cdot]$ satisfying antisymmetry and the Jacobi identity. It is:

simple if $\dim \mathfrak{g} > 1$ , $[\mathfrak{g},\mathfrak{g}] \neq 0$ , and it has no ideals other than $0$ and $\mathfrak{g}$ ;
semisimple if it is a direct sum of simple ideals, equivalently if its radical (maximal solvable ideal) is zero, equivalently (Cartan's criterion) if its Killing form is nondegenerate.

The Killing form is the symmetric bilinear form

1

\quad \kappa(X,Y) = \operatorname{tr}\big(\operatorname{ad}X \circ \operatorname{ad}Y\big), \qquad \operatorname{ad}X(Z)=[X,Z].

Derivation hint: $\operatorname{ad}$ is the adjoint representation $\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ ; composing two such operators and tracing gives an invariant bilinear form by the cyclicity of trace and the Jacobi identity ( $\kappa([X,Y],Z)=\kappa(X,[Y,Z])$ ).

DefinitionCartan Subalgebra and Root Space Decomposition

Let $\mathfrak{g}$ be semisimple over an algebraically closed field $k$ of characteristic $0$ . A Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ is a maximal abelian subalgebra acting diagonalizably under $\operatorname{ad}$ . Its common dimension $\ell = \dim\mathfrak{h}$ is the rank. Since the operators $\{\operatorname{ad}H : H\in\mathfrak{h}\}$ commute and are diagonalizable, $\mathfrak{g}$ decomposes into simultaneous eigenspaces:

2

\quad \mathfrak{g} = \mathfrak{h} \;\oplus\; \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, \qquad \mathfrak{g}_\alpha = \{X\in\mathfrak{g} : [H,X]=\alpha(H)\,X \ \ \forall H\in\mathfrak{h}\}.

Derivation hint: simultaneous diagonalization of the commuting family $\{\operatorname{ad}H\}$ groups $\mathfrak{g}$ by linear eigenvalue functionals $\alpha\in\mathfrak{h}^*$ ; the zero eigenspace is $\mathfrak{h}$ itself by maximality. The finite set $\Phi\subset\mathfrak{h}^*\setminus\{0\}$ of nonzero eigenfunctionals is the root system of $\mathfrak{g}$ . Each $\mathfrak{g}_\alpha$ is one-dimensional.

DefinitionAbstract Root System

A subset $\Phi$ of a Euclidean space $E$ (with inner product $(\cdot,\cdot)$ ) is a (reduced) root system if: 1. $\Phi$ is finite, spans $E$ , and $0\notin\Phi$ ; 2. for $\alpha\in\Phi$ , the only multiples of $\alpha$ in $\Phi$ are $\pm\alpha$ ; 3. for each $\alpha$ , the reflection $s_\alpha$ maps $\Phi$ to itself; 4. (integrality) for $\alpha,\beta\in\Phi$ , the Cartan integer

3

\quad \langle\beta,\alpha\rangle := \frac{2(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}, \qquad s_\alpha(\beta)=\beta-\langle\beta,\alpha\rangle\,\alpha.

Derivation hint: $\langle\beta,\alpha\rangle$ measures how $\beta$ projects onto $\alpha$ in units of $\alpha$ ; integrality comes from the eigenvalues of $\operatorname{ad}H_\alpha$ on the $\mathfrak{sl}_2$ -string through $\beta$ , which must be integers by representation theory of $\mathfrak{sl}_2$ . A root system is irreducible if $E$ does not decompose into orthogonal subspaces each spanned by a subset of $\Phi$ .

TheoremCartan–Killing Classification

Let $k$ be algebraically closed of characteristic $0$ . 1. The map $\mathfrak{g}\mapsto\Phi$ is a bijection between isomorphism classes of finite-dimensional simple Lie algebras over $k$ and isomorphism classes of irreducible reduced root systems. 2. Irreducible reduced root systems are classified by their connected Dynkin diagrams, and the complete list is:

4

\quad A_\ell\,(\ell\ge1),\ B_\ell\,(\ell\ge2),\ C_\ell\,(\ell\ge3),\ D_\ell\,(\ell\ge4),\ G_2,\ F_4,\ E_6,\ E_7,\ E_8.

Derivation hint: the four families are realized by $\mathfrak{sl}_{\ell+1},\mathfrak{so}_{2\ell+1},\mathfrak{sp}_{2\ell},\mathfrak{so}_{2\ell}$ ; the five exceptionals arise as the only remaining admissible Dynkin diagrams. The rank restrictions remove low-rank coincidences ( $A_1=B_1=C_1$ , $B_2=C_2$ , $D_3=A_3$ , $D_2=A_1\times A_1$ ).

Type	Lie algebra $\mathfrak{g}$	Rank $\ell$	$\dim\mathfrak{g}$	# roots
$A_\ell$	$\mathfrak{sl}_{\ell+1}$	$\ell$	$\ell(\ell+2)$	$\ell(\ell+1)$
$B_\ell$	$\mathfrak{so}_{2\ell+1}$	$\ell$	$\ell(2\ell+1)$	$2\ell^2$
$C_\ell$	$\mathfrak{sp}_{2\ell}$	$\ell$	$\ell(2\ell+1)$	$2\ell^2$
$D_\ell$	$\mathfrak{so}_{2\ell}$	$\ell$	$\ell(2\ell-1)$	$2\ell(\ell-1)$
$G_2$	—	$2$	$14$	$12$
$F_4$	—	$4$	$52$	$48$
$E_6$	—	$6$	$78$	$72$
$E_7$	—	$7$	$133$	$126$
$E_8$	—	$8$	$248$	$240$

ProofFrom a Simple Lie Algebra to a Dynkin Diagram and Back

Step 1 — Existence of Cartan subalgebra and root decomposition. Choose a regular semisimple element; the kernel of its $\operatorname{ad}$ generates a Cartan subalgebra $\mathfrak{h}$ . Since $\mathfrak{g}$ is semisimple, all of $\mathfrak{h}$ acts semisimply, giving the decomposition [2]. The Killing form restricted to $\mathfrak{h}$ is nondegenerate, transferring to a positive-definite form on the real span $E$ of $\Phi$ (after identifying $\mathfrak{h}^*$ via $\kappa$ ).

Step 2 — $\mathfrak{sl}_2$ -triples and integrality. For each $\alpha\in\Phi$ there exist $E_\alpha\in\mathfrak{g}_\alpha$ , $F_\alpha\in\mathfrak{g}_{-\alpha}$ , $H_\alpha\in\mathfrak{h}$ with

5

\quad [E_\alpha,F_\alpha]=H_\alpha,\quad [H_\alpha,E_\alpha]=2E_\alpha,\quad [H_\alpha,F_\alpha]=-2F_\alpha,

Derivation hint: $\mathfrak{g}_\alpha\oplus[\mathfrak{g}_\alpha,\mathfrak{g}_{-\alpha}]\oplus\mathfrak{g}_{-\alpha}$ closes into a copy of $\mathfrak{sl}_2$ ; normalize so the coroot $H_\alpha$ acts with eigenvalue $2$ on $E_\alpha$ . spanning a subalgebra $\cong\mathfrak{sl}_2(k)$ . The finite-dimensional representation theory of $\mathfrak{sl}_2$ forces every eigenvalue $\beta(H_\alpha)=\langle\beta,\alpha\rangle$ to be an integer — this is the source of property (4) and equation [3].

Step 3 — The root system is finite and crystallographic. The Weyl reflections $s_\alpha$ generate a finite group $W$ (the Weyl group) permuting $\Phi$ . Combining integrality with the Cauchy–Schwarz inequality gives the angle constraint

6

\quad \langle\alpha,\beta\rangle\,\langle\beta,\alpha\rangle = 4\cos^2\theta_{\alpha\beta}\in\{0,1,2,3\},

Derivation hint: the product of two Cartan integers equals $4\cos^2\theta$ ; since each factor is an integer and the product is $\le 4$ (strictly $<4$ for $\beta\ne\pm\alpha$ ), only the angles $90^\circ,120^\circ,135^\circ,150^\circ$ (and the corresponding length ratios $1,\sqrt2,\sqrt3$ ) can occur. so only finitely many relative configurations of two roots exist.

Step 4 — Simple roots and the Cartan matrix. Choose a hyperplane separating $\Phi$ into positive and negative roots; the indecomposable positive roots form a base (simple roots) $\Delta=\{\alpha_1,\dots,\alpha_\ell\}$ , a basis of $E$ . The Cartan matrix is

7

\quad A_{ij} = \langle\alpha_i,\alpha_j\rangle = \frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}, \qquad A_{ii}=2,\ A_{ij}\le 0\ (i\ne j),\ A_{ij}A_{ji}\in\{0,1,2,3\}.

Derivation hint: diagonal entries are $2$ by definition; off-diagonals are nonpositive integers because distinct simple roots form obtuse angles, a consequence of indecomposability.

Step 5 — Dynkin diagram and admissibility. Encode $A$ as a graph: one node per simple root, with $A_{ij}A_{ji}\in\{1,2,3\}$ edges joining nodes $i,j$ , and an arrow from the longer to the shorter root when lengths differ. The positive-definiteness of the symmetrized Cartan matrix forces the diagram to be one of: a chain (type $A$ ), a chain with one double bond at the end (types $B,C$ ), a chain with a fork (type $D$ ), the trivalent $E$ -series, $F_4$ , and $G_2$ . Any other configuration produces a non-positive-definite form (e.g. an affine diagram with determinant zero).

Step 6 — Reconstruction (Serre's theorem). Given any Cartan matrix $A$ from the list, the Lie algebra is recovered by generators $e_i,f_i,h_i$ subject to Serre relations:

8

\quad [h_i,h_j]=0,\ [e_i,f_j]=\delta_{ij}h_i,\ [h_i,e_j]=A_{ji}e_j,\ (\operatorname{ad}e_i)^{1-A_{ji}}e_j=0\ (i\ne j),

Derivation hint: the first three relations reproduce the $\mathfrak{sl}_2$ -triples and the $\mathfrak{h}$ -action; the last (Serre) relation truncates the root strings exactly as integrality demands, yielding a finite-dimensional algebra precisely when $A$ is of finite type. and the analogous relations for the $f_i$ . This proves the correspondence is a bijection: distinct diagrams give non-isomorphic algebras, and every admissible diagram is realized. $\blacksquare$

∎

Figure

The four connected Dynkin diagrams of low rank and the exceptional

G_2

. Nodes are simple roots; multiple bonds indicate length ratios.

mathematician · Élie Cartan | 1869–1951

Cartan, in his 1894 doctoral thesis Sur la structure des groupes de transformations finis et continus, took Killing's audacious but error-strewn classification and rebuilt it on rigorous foundations. He corrected Killing's mistaken belief in two distinct rank-4 exceptional algebras (they coincide as $F_4$ ), firmly established the five exceptional algebras, and pioneered the theory of weights and highest-weight representations. Over the following decades Cartan single-handedly created much of modern differential geometry — connections, moving frames, symmetric spaces, spinors — repeatedly returning to Lie theory as his structural backbone. His classification of real forms and of Riemannian symmetric spaces (1926) directly extends the work here.

🧠 Intuition

Think of a simple Lie algebra as a rigid crystal. The Cartan subalgebra is a maximal set of mutually commuting "measuring directions." Acting with these on the rest of the algebra splits it into one-dimensional pieces, each labeled by a vector (a root) in a Euclidean space. Because each pair of opposite roots generates an $\mathfrak{sl}_2$ — the smallest simple Lie algebra, the harmonic oscillator's spin algebra — the integrality of angular momentum quantum numbers ( $2j\in\mathbb{Z}$ ) is precisely what locks the geometry into a crystallographic pattern. Only a few crystal lattices are compatible with these quantization rules, and the Dynkin diagram is the symmetry "blueprint" of each. The reason the list is so short is the same reason there are only finitely many regular polytopes: rigidity from a single integrality constraint applied in every direction at once.

ExampleComputing the root system and Cartan matrix of

A_2 = \mathfrak{sl}_3

Given: $\mathfrak{g}=\mathfrak{sl}_3(\mathbb{C})$ , traceless $3\times3$ matrices, $\dim=8$ . Take $\mathfrak{h}=$ diagonal traceless matrices, $\operatorname{diag}(a_1,a_2,a_3)$ with $\sum a_i=0$ , so $\operatorname{rank}=2$ .

Step 1 — Roots. Let $\varepsilon_i(\operatorname{diag}(a_1,a_2,a_3))=a_i$ . The off-diagonal matrix unit $E_{ij}$ satisfies $[\,H,E_{ij}]=(a_i-a_j)E_{ij}$ , giving roots

\Phi=\{\varepsilon_i-\varepsilon_j : i\neq j\}, \quad |\Phi|=6.

Step 2 — Simple roots. Choose $\alpha_1=\varepsilon_1-\varepsilon_2$ , $\alpha_2=\varepsilon_2-\varepsilon_3$ . All positive roots: $\alpha_1,\alpha_2,\alpha_1+\alpha_2$ .

Step 3 — Inner products. With $(\varepsilon_i,\varepsilon_j)=\delta_{ij}$ : $(\alpha_1,\alpha_1)=(\alpha_2,\alpha_2)=2$ , $(\alpha_1,\alpha_2)=-1$ .

Step 4 — Cartan matrix via [7]:

A=\begin{pmatrix}2 & -1\\ -1 & 2\end{pmatrix}.

Angle: $\cos\theta=(\alpha_1,\alpha_2)/2=-1/2\Rightarrow\theta=120^\circ$ . Single bond, equal lengths.

Boxed answer:

\boxed{\ \Phi \cong A_2,\quad \dim\mathfrak{sl}_3=8,\ \text{a hexagon of 6 roots, single bond, Weyl group}\ S_3\ (\text{order }6).\ }

ExampleVerifying the dimension count of

E_8

from its diagram

Given: $E_8$ has rank $\ell=8$ and (by the classification) $240$ roots.

Step 1 — Dimension formula. A simple Lie algebra satisfies $\dim\mathfrak{g}=\ell+|\Phi|$ , since $\mathfrak{h}$ contributes $\ell$ and each root contributes a one-dimensional space.

Step 2 — Apply.

\dim E_8 = 8 + 240 = 248.

Step 3 — Cross-check via root lengths. All $E_8$ roots have squared length $2$ in the normalized $E_8$ lattice; the $240$ minimal vectors of the $E_8$ lattice (kissing number in $8$ dimensions) reproduce $|\Phi|=240$ .

Step 4 — Coxeter number. $h=30$ , dual Coxeter number $h^\vee=30$ (self-dual), and indeed $\ell\cdot h = 8\cdot 30 = 240 = |\Phi|$ , a general identity.

Boxed answer:

\boxed{\ \dim E_8 = 248,\quad |\Phi|=240,\quad h=h^\vee=30.\ }

⚠️ Common mistake

" $B_\ell$ and $C_\ell$ are isomorphic because they have the same dimension." False. $\mathfrak{so}_{2\ell+1}$ and $\mathfrak{sp}_{2\ell}$ both have dimension $\ell(2\ell+1)$ , but their Dynkin diagrams differ: the double-bond arrow points toward the short root in $B_\ell$ and toward the long root in $C_\ell$ . Their root systems are dual ( $B_\ell^\vee = C_\ell$ ), not equal, and the algebras are non-isomorphic for $\ell\ge 3$ . They genuinely coincide only at $\ell=2$ ( $B_2=C_2$ ).

⚠️ Common mistake

The classification fails outside its hypotheses. Over fields of positive characteristic there are additional simple Lie algebras (Cartan-type, Witt, Hamiltonian algebras) not in this list. Over non-algebraically-closed fields one must classify real forms (each complex type has several), and in infinite dimensions one obtains the much larger world of Kac–Moody and affine Lie algebras.

Counterexample

Low-rank coincidences violate naïve "one diagram per algebra" expectations. The seemingly distinct diagrams $D_2$ and $A_1\times A_1$ describe the same (non-simple, semisimple) algebra: $\mathfrak{so}_4\cong\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ . Likewise $D_3=A_3$ ( $\mathfrak{so}_6\cong\mathfrak{sl}_4$ ) and $B_2=C_2$ ( $\mathfrak{so}_5\cong\mathfrak{sp}_4$ ). These are exactly why the rank ranges in [4] are restricted — to make the parametrization a bijection with no double-counting.

connection

The classification sits at the crossroads of several theories. Through Root Systems and the Weyl Group it connects to reflection-group combinatorics and Coxeter theory. Via Highest Weight Theory it controls all finite-dimensional Representations of Semisimple Lie Algebras. Exponentiating, it underlies the classification of Compact Lie Groups and, through Cartan's Symmetric Spaces, of Riemannian symmetric spaces. The same Cartan matrices generalize to define Kac-Moody Algebras (drop positive-definiteness) and, with quantum deformation, Quantum Groups. The exceptional $E_8$ links to the Leech Lattice and modular forms via lattice theory.

🔬 Where this appears

⏳ History

- 1873–1888 — Sophus Lie founds the theory of continuous transformation groups and their infinitesimal algebras.

- 1888–1890 — Wilhelm Killing publishes Die Zusammensetzung der stetigen endlichen Transformationsgruppen, introducing the characteristic equation, roots, rank, and the Cartan matrix concept; he obtains essentially the full list but with gaps and the erroneous duplicate exceptional algebra.

- 1894 — Élie Cartan's thesis rigorously establishes the classification, corrects Killing's errors (merging the spurious pair into

F_4

), and definitively names the four families and five exceptionals.

- 1913 — Cartan develops the theory of weights and classifies irreducible representations via highest weights.

- 1925–1926 — Hermann Weyl proves complete reducibility (Weyl's theorem) and the character/dimension formulas, recasting the subject in terms of compact groups (the "unitarian trick").

- 1933 — Bartel van der Waerden gives a purely algebraic proof of the classification using root systems abstractly.

- 1934 — Ernst Witt independently formalizes root systems and their classification.

- 1947 — Eugene Dynkin introduces the Dynkin diagram, making the enumeration transparent and standard; Serre later (1966) supplies the generators-and-relations reconstruction.

- 1950s onward — Chevalley constructs the algebras over arbitrary fields (Chevalley groups), extending the reach of the classification to finite simple groups of Lie type.

open-problem

The finite-dimensional classification is complete and closed — there are no open cases. The frontier lies in its generalizations and refinements. The classification of infinite-dimensional Lie algebras (Kac–Moody of indefinite type) remains wide open: only finite and affine types are fully understood, while the hyperbolic and general indefinite Cartan matrices resist classification. The structure and representation theory of these (e.g. the conjectural role of $E_{10}$ and $E_{11}$ in M-theory and cosmological billiards) is an active research domain. In positive characteristic, the classification of simple modular Lie algebras was only completed for $p>3$ (Block–Wilson–Premet–Strade, c. 2004), and the cases $p=2,3$ involve exotic algebras still being catalogued. Whether the deepest physical role of $E_8$ — proposed in various unification and "theory of everything" schemes — is fundamental or coincidental remains an open question of physics rather than mathematics.

The Cartan Classification of Simple Lie Algebras

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also