Root-space decomposition

Intuition [Beginner]

A root-space decomposition is a way of splitting a semisimple Lie algebra into a "centre" part and a collection of "direction" parts, each labelled by a vector called a root. The centre part is the Cartan subalgebra: a maximal commuting subalgebra of diagonal-type elements. The direction parts are one-dimensional spaces, each carrying a specific charge relative to the Cartan subalgebra.

The word "root" comes from the eigenvalues. If you take an element $H$ of the Cartan subalgebra and let it act on the full Lie algebra, the action decomposes into eigenspaces. Each eigenspace corresponds to a specific eigenvalue, and the eigenvalue is a linear function of $H$. These linear functions are the roots.

The roots are not random: they form a highly symmetric geometric pattern. For ${\mathfrak{sl}}_3\mathbb{C}$, there are six roots, pointing from the centre of a regular hexagon to its six vertices. This geometric regularity is what makes the classification of semisimple Lie algebras possible: the root system determines the Lie algebra completely.

Visual [Beginner]

A picture of the root system $A_2$ (for ${\mathfrak{sl}}_3\mathbb{C}$): six arrows radiating from the origin in a hexagonal pattern. The arrows come in three opposite pairs, labelled by the roots $\alpha ,-\alpha$, $\beta ,-\beta$, and $\alpha +\beta ,-(\alpha +\beta )$. Two short arrows at the origin represent the two-dimensional Cartan subalgebra. Each arrow is a root space: a one-dimensional subspace of the Lie algebra.

The picture captures the fundamental geometry: the Lie algebra is the Cartan subalgebra (the origin) plus one direction for each root (each arrow), and the roots form a symmetric pattern that determines the Lie algebra.

Worked example [Beginner]

Consider ${\mathfrak{sl}}_3\mathbb{C}$, the Lie algebra of traceless $3\times 3$ matrices. This algebra has dimension $8$. The Cartan subalgebra $\mathfrak{h}$ consists of diagonal traceless matrices, of dimension $2$.

Step 1. A basis for $\mathfrak{h}$ is $H_1=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right)$ and $H_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right)$. These commute: $[H_1,H_2]=0$.

Step 2. The off-diagonal basis elements are the six matrices $E_{ij}$ (one at position $(i,j)$, zero elsewhere) for $i\ne j$. Each satisfies $[H,E_{ij}]=(h_i-h_j)E_{ij}$ where $h_k$ is the $k$-th diagonal entry of $H$.

Step 3. The six roots are the linear functions $\alpha (H)=h_1-h_2$, $\beta (H)=h_2-h_3$, $\alpha +\beta (H)=h_1-h_3$, and their negatives. The root spaces are ${\mathfrak{g}}_{\alpha }=\mathbb{C}{E}_{12}$, ${\mathfrak{g}}_{\beta }=\mathbb{C}{E}_{23}$, ${\mathfrak{g}}_{\alpha +\beta }=\mathbb{C}{E}_{13}$, and their negatives.

What this tells us: ${\mathfrak{sl}}_3\mathbb{C}=\mathfrak{h}\oplus {\mathfrak{g}}_{\alpha }\oplus {\mathfrak{g}}_{-\alpha }\oplus {\mathfrak{g}}_{\beta }\oplus {\mathfrak{g}}_{-\beta }\oplus {\mathfrak{g}}_{\alpha +\beta }\oplus {\mathfrak{g}}_{-(\alpha +\beta )}$, a direct sum of $8$ one-dimensional spaces (two from $\mathfrak{h}$, six from the roots), with each root space labelled by its eigenvalue function.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra $\mathfrak{h}$. For a linear functional $\alpha \in {\mathfrak{h}}^{\ast }$, define the root space $$ \mathfrak{g}_\alpha = {X \in \mathfrak{g} : [H, X] = \alpha(H) X \text{ for all } H \in \mathfrak{h}}. $$

A root is a nonzero $\alpha \in {\mathfrak{h}}^{\ast }$ for which ${\mathfrak{g}}_{\alpha }\ne 0$. The set of all roots is denoted $\Phi \subset {\mathfrak{h}}^{\ast }$.

Theorem (Root-space decomposition). The Lie algebra $\mathfrak{g}$ decomposes as a direct sum of vector spaces: $$ \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha. $$

Each root space ${\mathfrak{g}}_{\alpha }$ is one-dimensional. The set of roots $\Phi$ spans ${\mathfrak{h}}^{\ast }$. The brackets satisfy $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subseteq {\mathfrak{g}}_{\alpha +\beta }$ when $\alpha +\beta \in \Phi$, and $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]=0$ when $\alpha +\beta \ne 0$ and $\alpha +\beta \notin \Phi$.

Counterexamples to common slips

• Not every linear functional is a root. The root spaces are eigenspaces of the commuting family $\{{\mathrm{ad}}_H:H\in \mathfrak{h}\}$. Most linear functionals give zero eigenspaces; only finitely many give nonzero eigenspaces. For ${\mathfrak{sl}}_3$, the space ${\mathfrak{h}}^{\ast }$ is two-dimensional (infinitely many functionals), but only six of them are roots.
• The Cartan subalgebra is not a root space. The decomposition separates $\mathfrak{h}$ (the zero-eigenspace) from the root spaces (nonzero eigenvalues). The notation ${\mathfrak{g}}_0=\mathfrak{h}$ is sometimes used, but $0$ is not called a root.
• Root spaces can be one-dimensional even when the root system looks two-dimensional. The root system is a subset of ${\mathfrak{h}}^{\ast }$, not of $\mathfrak{g}$. Each root $\alpha$ is a point in ${\mathfrak{h}}^{\ast }$, and ${\mathfrak{g}}_{\alpha }$ is a single line in $\mathfrak{g}$.

Key theorem with proof [Intermediate+]

Theorem (Root-space decomposition of a semisimple Lie algebra). Let $\mathfrak{g}$ be a semisimple complex Lie algebra with Cartan subalgebra $\mathfrak{h}$. Then:

(a) $\mathfrak{g}=\mathfrak{h}\oplus {⨁}_{\alpha \in \Phi }{\mathfrak{g}}_{\alpha }$ (direct sum of vector spaces).

(b) $\dim {\mathfrak{g}}_{\alpha }=1$ for each $\alpha \in \Phi$.

(c) $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subseteq {\mathfrak{g}}_{\alpha +\beta }$ for all $\alpha ,\beta \in \Phi \cup \{0\}$.

(d) $\Phi$ spans $\mathfrak{h}^$and$\alpha \in \Phi \Rightarrow -\alpha \in \Phi$.*

Proof of (a). Since $\mathfrak{h}$ is abelian and consists of semisimple elements (by the semisimplicity of $\mathfrak{g}$), the operators $\{{\mathrm{ad}}_H:H\in \mathfrak{h}\}$ form a commuting family of diagonalisable endomorphisms of $\mathfrak{g}$. A commuting family of diagonalisable operators can be simultaneously diagonalised. The simultaneous eigenspace decomposition of $\mathfrak{g}$ under $\mathfrak{h}$ is: $$ \mathfrak{g} = \bigoplus_{\alpha \in \mathfrak{h}^*} \mathfrak{g}\alpha $$ where $\mathfrak{g}\alpha = {X \in \mathfrak{g} : [H, X] = \alpha(H) X \text{ for all } H \in \mathfrak{h}}$andthedirectsumisoverall$\alpha$with$\mathfrak{g}\alpha \neq 0$.Thezero-weightspace$\mathfrak{g}0$isthecentraliser$C\mathfrak{g}(\mathfrak{h})$,whichequals$\mathfrak{h}$(byProposition1of[\mathrm{07.06.17}]).Setting$\Phi = {\alpha \neq 0 : \mathfrak{g}\alpha \neq 0}$ gives (a).

Proof of (b). Let $\alpha \in \Phi$. The root space ${\mathfrak{g}}_{\alpha }$ is one-dimensional. Suppose $\dim {\mathfrak{g}}_{\alpha }\ge 2$. The space ${⨁}_{n\ge 0}{\mathfrak{g}}_{n\alpha }$ is a subspace of $\mathfrak{g}$ invariant under $\mathrm{ad}(\mathfrak{h}+{\mathfrak{g}}_{\alpha })$. The operator ${\mathrm{ad}}_{E_{\alpha }}$ for a nonzero $E_{\alpha }\in {\mathfrak{g}}_{\alpha }$ raises the $\alpha$-weight by $\alpha$. Since $\dim \mathfrak{g}$ is finite, only finitely many $n\alpha$ can be roots. The Killing form gives $B({\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta })=0$ when $\alpha +\beta \ne 0$ (by invariance: $B([H,X],Y)=\alpha (H)B(X,Y)=-\beta (H)B(X,Y)$, so $(\alpha (H)+\beta (H))B(X,Y)=0$ for all $H$; if $\alpha +\beta \ne 0$, choose $H$ with $(\alpha +\beta )(H)\ne 0$ to conclude $B(X,Y)=0$). The Killing form pairs ${\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{-\alpha }$ non-degenerately (since $B$ is non-degenerate and all other pairings vanish). Both ${\mathfrak{g}}_{\alpha }$ and ${\mathfrak{g}}_{-\alpha }$ must be nonzero and dual under $B$. If $\dim {\mathfrak{g}}_{\alpha }\ge 2$, the subspace ${\mathfrak{g}}_{\alpha }\oplus {\mathfrak{g}}_{-\alpha }\oplus [{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha }]$ would have dimension at least $5$, but the ${\mathfrak{sl}}_2$-subalgebra generated by $E_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $F_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ has dimension $3$ (spanned by $E_{\alpha },F_{\alpha },H_{\alpha }=[E_{\alpha },F_{\alpha }]$), and the representation theory of ${\mathfrak{sl}}_2$ forces each weight space to be one-dimensional in this subalgebra. So $\dim {\mathfrak{g}}_{\alpha }=1$.

Proof of (c). For $X\in {\mathfrak{g}}_{\alpha }$, $Y\in {\mathfrak{g}}_{\beta }$, and $H\in \mathfrak{h}$: $[H,[X,Y]]=[[H,X],Y]+[X,[H,Y]]=\alpha (H)[X,Y]+\beta (H)[X,Y]=(\alpha +\beta )(H)[X,Y]$. So $[X,Y]\in {\mathfrak{g}}_{\alpha +\beta }$.

Proof of (d). If $\Phi$ did not span ${\mathfrak{h}}^{\ast }$, there would exist nonzero $H_0\in \mathfrak{h}$ with $\alpha (H_0)=0$ for all $\alpha \in \Phi$. Then ${\mathrm{ad}}_{H_0}$ acts as zero on every ${\mathfrak{g}}_{\alpha }$ and on $\mathfrak{h}$, so ${\mathrm{ad}}_{H_0}=0$, meaning $H_0\in Z(\mathfrak{g})=0$, a contradiction. For the symmetry: since $B$ pairs ${\mathfrak{g}}_{\alpha }$ non-degenerately with ${\mathfrak{g}}_{-\alpha }$, both must be nonzero, so $\alpha \in \Phi \Rightarrow -\alpha \in \Phi$. $\square$

Bridge. The root-space decomposition builds toward 07.06.03 where the root system is axiomatised and classified, and appears again in 07.06.05 where the Dynkin diagram encodes the root system in a combinatorial graph. The foundational reason the decomposition works is that the Cartan subalgebra consists of simultaneously diagonalisable operators whose eigenspaces are one-dimensional — this is exactly the content of the semisimplicity of the elements in $\mathfrak{h}$ combined with the non-degeneracy of the Killing form. The central insight is that the roots carry the complete bracket structure: $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subseteq {\mathfrak{g}}_{\alpha +\beta }$ means the Lie algebra is determined by the root system plus a set of structure constants. The bridge is between the abstract Lie algebra and the combinatorial root system, and putting these together generalises to the full Cartan-Killing classification: every semisimple Lie algebra is determined up to isomorphism by its root system.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Root system axioms). The set $\Phi$ of roots of a semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, viewed in the Euclidean space $E={\mathfrak{h}}_{\mathbb{R}}^{\ast }$ with inner product induced by the Killing form, satisfies: (R1) $\Phi$ is finite, spans $E$, and $0\notin \Phi$; (R2) $\alpha \in \Phi \Rightarrow -\alpha \in \Phi$; (R3) for each $\alpha \in \Phi$, the reflection $s_{\alpha }(\beta )=\beta -⟨\beta ,{\alpha }^{\vee }⟩\alpha$ maps $\Phi$ to $\Phi$; (R4) $⟨\beta ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ for all $\alpha ,\beta \in \Phi$.

The integers $⟨\beta ,{\alpha }^{\vee }⟩=2⟨\beta ,\alpha ⟩/⟨\alpha ,\alpha ⟩$ are called the Cartan integers. Their possible values are $\{0,\pm 1,\pm 2,\pm 3\}$, with the constraint that if $⟨\alpha ,\beta ⟩>0$ then $\alpha -\beta \in \Phi$.

Theorem 2 (The Weyl group). The Weyl group $W$ of the root system $\Phi$ is the group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. The Weyl group is finite, acts on $\Phi$ by permutations, and acts on $\mathfrak{h}$ (via the Killing-form identification $\mathfrak{h}\cong {\mathfrak{h}}^{\ast }$). For type $A_n$, $W\cong S_{n+1}$ (the symmetric group). The order of $W$ for each type is given by $|W(A_n)|=(n+1)!$, $|W(B_n)|=2^{n}n!$, $|W(D_n)|=2^{n-1}n!$.

Theorem 3 (Simple roots and the Weyl chamber). A simple system $\Delta =\{{\alpha }_1,\dots ,{\alpha }_r\}$ is a subset of $\Phi$ such that every root is a linear combination of simple roots with all coefficients either non-negative or all non-positive. The simple roots form a basis of ${\mathfrak{h}}^{\ast }$. The connected component of ${\mathfrak{h}}_{\mathbb{R}}^{\ast }\setminus {\bigcup }_{\alpha }\ker (\alpha )$ containing the half-sum of positive roots $\rho$ is the fundamental Weyl chamber.

Theorem 4 (Dynkin diagrams). The Dynkin diagram of a root system $\Phi$ with simple system $\Delta$ is a graph with one vertex for each simple root and edges determined by the Cartan integers between pairs of simple roots. The irreducible Dynkin diagrams are classified into four infinite families ($A_n$, $B_n$, $C_n$, $D_n$) and five exceptional types ($E_6$, $E_7$, $E_8$, $F_4$, $G_2$).

Theorem 5 (Serre relations). A semisimple Lie algebra with root system $\Phi$ and simple roots $\Delta =\{{\alpha }_1,\dots ,{\alpha }_r\}$ is generated by $\{E_i,F_i,H_i:i=1,\dots ,r\}$ subject to the Chevalley-Serre relations: $[H_i,H_j]=0$, $[H_i,E_j]=⟨{\alpha }_j,{\alpha }_i^{\vee }⟩E_j$, $[H_i,F_j]=-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩F_j$, $[E_i,F_j]={\delta }_{ij}{H}_i$, $({\mathrm{ad}}_{E_i}{)}^{1-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩}(E_j)=0$, and $({\mathrm{ad}}_{F_i}{)}^{1-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩}(F_j)=0$.

Theorem 6 (Classification theorem). Two semisimple complex Lie algebras are isomorphic if and only if their root systems are isomorphic. The root systems are classified by the Dynkin diagrams, completing the classification of simple complex Lie algebras.

Theorem 7 (Strings). For roots $\alpha ,\beta \in \Phi$ with $\beta \ne \pm \alpha$, the $\alpha$-string through $\beta$ is the set $\{\beta +n\alpha :n\in \mathbb{Z}\}\cap \Phi$. It is an unbroken string $\beta -p\alpha ,\dots ,\beta ,\dots ,\beta +q\alpha$ with $p-q=⟨\beta ,{\alpha }^{\vee }⟩$.

Synthesis. The foundational reason the root-space decomposition works is that the Cartan subalgebra consists of commuting semisimple operators whose simultaneous eigenspaces are one-dimensional, and the central insight is that the roots form a finite set of vectors in a Euclidean space satisfying the four root-system axioms. Putting these together with the Killing form, the root system inherits an inner product that makes it a crystallographic reflection group, and the bridge is between the Lie algebra structure (brackets, Killing form, Cartan subalgebra) and the combinatorial root system (reflection group, Dynkin diagram, Cartan matrix). This is exactly the structure that identifies each semisimple Lie algebra with a unique root system: the Serre relations reconstruct the Lie algebra from the Dynkin diagram, and the classification theorem confirms the correspondence is bijective. The pattern recurs in 07.06.04 where the Weyl group acts on the root system, in 07.06.05 where the Dynkin diagram encodes the root system as a graph, and in 07.06.07 where the Weyl character formula expresses characters in terms of the Weyl group acting on weights. The generalises direction runs from the root-space decomposition of individual Lie algebras to the full classification of all semisimple Lie algebras via the Killing-Cartan programme.

Full proof set [Master]

Proposition 1 (Killing form pairs ${\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{-\alpha }$). The Killing form $B$ pairs ${\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{-\alpha }$ non-degenerately for each $\alpha \in \Phi$: the map ${\mathfrak{g}}_{\alpha }\to {\mathfrak{g}}_{-\alpha }^{\ast }$ given by $X\mapsto B(X,\cdot )$ is an isomorphism.

Proof. By Exercise 4, $B({\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta })=0$ when $\alpha +\beta \ne 0$. So the only nonzero pairings are $B({\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha })$ for each $\alpha \in \Phi$, and $B(\mathfrak{h},\mathfrak{h})$. Since $B$ is non-degenerate on $\mathfrak{g}$ (Cartan's criterion 07.06.16) and is block-diagonal with respect to the decomposition $\mathfrak{g}=\mathfrak{h}\oplus {⨁}_{\alpha \in \Phi }{\mathfrak{g}}_{\alpha }$, the restriction $B:{\mathfrak{g}}_{\alpha }\times {\mathfrak{g}}_{-\alpha }\to \mathbb{C}$ must be non-degenerate. Otherwise there would exist $0\ne X\in {\mathfrak{g}}_{\alpha }$ with $B(X,Y)=0$ for all $Y\in {\mathfrak{g}}_{-\alpha }$ and all $Y$ in other root spaces and $\mathfrak{h}$ (which pair to zero with ${\mathfrak{g}}_{\alpha }$ anyway), contradicting non-degeneracy of $B$. $\square$

Proposition 2 (Root strings are unbroken). For $\alpha ,\beta \in \Phi$ with $\beta \ne \pm \alpha$, the $\alpha$-string through $\beta$ is an unbroken sequence: if $\beta +p\alpha$ and $\beta +q\alpha$ are roots with $p\le q$, then $\beta +n\alpha$ is a root for all $p\le n\le q$.

Proof. Consider the subspace $V={⨁}_{n\in \mathbb{Z}}{\mathfrak{g}}_{\beta +n\alpha }$. This is an ${\mathfrak{sl}}_2$-module under the triple $\{E_{\alpha },H_{\alpha },F_{\alpha }\}$ where $E_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $F_{\alpha }\in {\mathfrak{g}}_{-\alpha }$. The element $H_{\alpha }$ acts on ${\mathfrak{g}}_{\beta +n\alpha }$ by the scalar $(\beta +n\alpha )(H_{\alpha })=⟨\beta ,{\alpha }^{\vee }⟩+2n$. Any finite-dimensional ${\mathfrak{sl}}_2$-module is a direct sum of irreducible modules, each with weights forming an unbroken string $\{m,m-2,\dots ,-(m-2),-m\}$ for some non-negative integer $m$. Since each ${\mathfrak{g}}_{\beta +n\alpha }$ is one-dimensional (or zero), the ${\mathfrak{sl}}_2$-weights on $V$ are distinct. An irreducible ${\mathfrak{sl}}_2$-module with distinct weights (each appearing once) has weights forming an unbroken string. So the roots $\beta +n\alpha$ that appear form an unbroken string. $\square$

Proposition 3 (Integrality of Cartan integers). For $\alpha ,\beta \in \Phi$, the Cartan integer $⟨\beta ,{\alpha }^{\vee }⟩=2⟨\beta ,\alpha ⟩/⟨\alpha ,\alpha ⟩$ is an integer.

Proof. The $\alpha$-string through $\beta$ has the form $\beta -p\alpha ,\dots ,\beta +q\alpha$ where $p,q\ge 0$. The ${\mathfrak{sl}}_2$-weights on this string are $⟨\beta ,{\alpha }^{\vee }⟩+2n$ for $n=-p,\dots ,q$. The highest weight is $⟨\beta ,{\alpha }^{\vee }⟩+2q$ and the lowest is $⟨\beta ,{\alpha }^{\vee }⟩-2p$. By the ${\mathfrak{sl}}_2$ weight symmetry, the highest and lowest weights are negatives of each other: $⟨\beta ,{\alpha }^{\vee }⟩+2q=-(⟨\beta ,{\alpha }^{\vee }⟩-2p)$. So $2⟨\beta ,{\alpha }^{\vee }⟩=2(p-q)$, giving $⟨\beta ,{\alpha }^{\vee }⟩=p-q\in \mathbb{Z}$. $\square$

Connections [Master]

• Root system 07.06.03. The root-space decomposition produces the root system $\Phi$ as a subset of ${\mathfrak{h}}^{\ast }$, and 07.06.03 axiomatises the root system independently of any Lie algebra. The root-space decomposition proves that the roots of a semisimple Lie algebra satisfy the root-system axioms; conversely, the Serre relations construct a Lie algebra from any root system. The two units together establish the bijection between semisimple Lie algebras and root systems.

• Cartan's criterion 07.06.16. The non-degeneracy of the Killing form (Cartan's criterion for semisimplicity) is used throughout the root-space decomposition: it identifies $\mathfrak{h}\cong {\mathfrak{h}}^{\ast }$ via $B$, pairs ${\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{-\alpha }$, and provides the Euclidean inner product on the root lattice. Without Cartan's criterion, the root system does not inherit an inner product and the axioms (R3) and (R4) cannot be formulated.

• Casimir element 07.06.10. The Casimir element $C={\sum }_a{X}_a{X}^a$ is expressed in the root-space-adapted basis $\{H_i,E_{\alpha }\}$ with dual basis $\{H^i,E_{-\alpha }\}$ as $C={\sum }_i{H}_i{H}^i+{\sum }_{\alpha \in {\Phi }^{+}}(E_{\alpha }{E}_{-\alpha }+E_{-\alpha }{E}_{\alpha })$. The root-space decomposition is the structural input that makes the Casimir computation explicit, and the Casimir eigenvalue formula $c_{V_{\lambda }}=⟨\lambda ,\lambda +2\rho ⟩$ is expressed in terms of the root-system inner product.

Historical & philosophical context [Master]

Wilhelm Killing introduced the root-space decomposition in his four-part paper Die Zusammensetzung der stetigen endlichen Transformationsgruppen published in Mathematische Annalen (volumes 31, 33, 34, 36, 1888–1890) ^{[Killing1888]}. Killing's approach was computational and sometimes imprecise — he identified the root systems of the classical Lie algebras and the exceptional types but made errors in the details of the root classification. The root system was for Killing a tool for classifying transformation groups, not an independent mathematical object.

Élie Cartan corrected and completed Killing's classification in his 1894 Paris thesis ^[Cartan1894], giving rigorous proofs of the root-space decomposition for all simple types. Cartan's thesis is the origin of the modern presentation: the Cartan subalgebra, the root spaces, and the root system as a structural invariant.

The axiomatisation of root systems as independent combinatorial objects (the four axioms R1–R4) is due to Cartan's later work in the 1920s and was crystallised by Hermann Weyl in his 1925–26 four-part paper Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen in Math. Z. ^[pending]. The Dynkin diagram encoding was introduced by Eugene Dynkin in 1947 ^[Dynkin1947] as a compact way to record the Cartan matrix. The Serre relations, reconstructing the Lie algebra from the Dynkin diagram, were proved by Jean-Pierre Serre in the early 1960s and appear in his Complex Semisimple Lie Algebras (1966).

Bibliography [Master]

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  pages   = {252--290}
}

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}

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}

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