[proofplan] We attach to a finite-dimensional semisimple Lie algebra $\mathfrak{g}$ over the algebraically closed characteristic-zero field $k$ the Dynkin diagram of its root system relative to a Cartan subalgebra. Conjugacy of Cartan subalgebras and the Weyl-group conjugacy of simple systems make this diagram independent of choices. The classification of irreducible finite reduced crystallographic root systems gives the listed connected diagrams, while Serre's existence and uniqueness theorem for finite-type Cartan matrices constructs and identifies the corresponding simple Lie algebras. Finally, direct sums of simple ideals correspond exactly to disjoint unions of connected diagrams. [/proofplan]

[step:Attach a Dynkin diagram to a semisimple Lie algebra]Let $k$ denote the algebraically closed field of characteristic $0$, and let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $k$. By the Cartan-subalgebra existence theorem for finite-dimensional semisimple Lie algebras over algebraically closed fields of characteristic $0$, choose a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$. Define the root space attached to a nonzero linear functional $\alpha \in \mathfrak{h}^*$ by \begin{align*} \mathfrak{g}_\alpha := \{x \in \mathfrak{g} : [h,x] = \alpha(h)x \text{ for every } h \in \mathfrak{h}\}. \end{align*} Let \begin{align*} \Phi(\mathfrak{g},\mathfrak{h}) := \{\alpha \in \mathfrak{h}^* \setminus \{0\} : \mathfrak{g}_\alpha \neq 0\}. \end{align*} The root-space decomposition theorem gives \begin{align*} \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi(\mathfrak{g},\mathfrak{h})} \mathfrak{g}_\alpha, \end{align*} and asserts that $\Phi(\mathfrak{g},\mathfrak{h})$ is a finite reduced crystallographic root system in the real [vector space](/page/Vector%20Space) spanned by $\Phi(\mathfrak{g},\mathfrak{h})$. Choose a simple system $\Delta \subset \Phi(\mathfrak{g},\mathfrak{h})$. The associated Cartan matrix is the integer matrix \begin{align*} A_\Delta = (a_{\alpha\beta})_{\alpha,\beta \in \Delta}, \qquad a_{\alpha\beta} := \frac{2(\alpha,\beta)}{(\beta,\beta)}, \end{align*} where $(\cdot,\cdot)$ is the Weyl-invariant Euclidean inner product on the real span of the roots. The Dynkin diagram $D(\mathfrak{g},\mathfrak{h},\Delta)$ is the graph encoded by this Cartan matrix.[/step]

[guided]We first turn a Lie algebra into a combinatorial object. Let $k$ be the algebraically closed field of characteristic $0$, and let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $k$. The Cartan-subalgebra existence theorem applies precisely in this setting: [finite dimensionality](/theorems/1534), semisimplicity, algebraic closedness, and characteristic $0$ are the hypotheses needed for the standard root-space theory. Choose a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$. For each nonzero linear functional $\alpha \in \mathfrak{h}^*$, define \begin{align*} \mathfrak{g}_\alpha := \{x \in \mathfrak{g} : [h,x] = \alpha(h)x \text{ for every } h \in \mathfrak{h}\}. \end{align*} This is the simultaneous eigenspace for the adjoint action of $\mathfrak{h}$. Define the root system by \begin{align*} \Phi(\mathfrak{g},\mathfrak{h}) := \{\alpha \in \mathfrak{h}^* \setminus \{0\} : \mathfrak{g}_\alpha \neq 0\}. \end{align*} The root-space decomposition theorem gives \begin{align*} \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi(\mathfrak{g},\mathfrak{h})} \mathfrak{g}_\alpha. \end{align*} It also states that $\Phi(\mathfrak{g},\mathfrak{h})$ is a finite reduced crystallographic root system. This is the point where the Lie bracket is converted into finite Euclidean geometry. Choose a simple system $\Delta \subset \Phi(\mathfrak{g},\mathfrak{h})$. For $\alpha,\beta \in \Delta$, define \begin{align*} a_{\alpha\beta} := \frac{2(\alpha,\beta)}{(\beta,\beta)}, \end{align*} where $(\cdot,\cdot)$ is the Weyl-invariant Euclidean inner product on the real span of the roots. The integers $a_{\alpha\beta}$ form the Cartan matrix $A_\Delta = (a_{\alpha\beta})_{\alpha,\beta \in \Delta}$, and the Dynkin diagram $D(\mathfrak{g},\mathfrak{h},\Delta)$ is the graph encoded by this matrix.[/guided]

[step:Check that the diagram is independent of the choices]If $\mathfrak{h}_1$ and $\mathfrak{h}_2$ are Cartan subalgebras of $\mathfrak{g}$, the [conjugacy theorem for Cartan subalgebras](/theorems/4681) gives an automorphism $\varphi: \mathfrak{g} \to \mathfrak{g}$ with $\varphi(\mathfrak{h}_1)=\mathfrak{h}_2$. The induced linear isomorphism $\mathfrak{h}_2^* \to \mathfrak{h}_1^*$ identifies the two root systems and preserves Cartan integers. Within a fixed root system, any two simple systems are conjugate under the Weyl group, and Weyl-group elements preserve the Cartan matrix up to simultaneous relabelling of the simple roots. Hence $D(\mathfrak{g},\mathfrak{h},\Delta)$ depends only on the isomorphism class of $\mathfrak{g}$; write it as $D(\mathfrak{g})$.[/step]

[guided]There are two choices in the construction: the Cartan subalgebra and the simple roots. We verify that neither affects the final diagram. If $\mathfrak{h}_1$ and $\mathfrak{h}_2$ are Cartan subalgebras of the same finite-dimensional semisimple Lie algebra $\mathfrak{g}$ over $k$, the Cartan-subalgebra conjugacy theorem applies under the same hypotheses already in force: $k$ is algebraically closed of characteristic $0$, and $\mathfrak{g}$ is finite-dimensional and semisimple. Therefore there is a Lie algebra automorphism $\varphi: \mathfrak{g} \to \mathfrak{g}$ such that $\varphi(\mathfrak{h}_1)=\mathfrak{h}_2$. This automorphism carries root spaces for $\mathfrak{h}_1$ to root spaces for $\mathfrak{h}_2$. Consequently it identifies the two root systems and preserves the Cartan integers \begin{align*} \frac{2(\alpha,\beta)}{(\beta,\beta)}. \end{align*} Now fix one root system. The theorem that simple systems are conjugate under the Weyl group says that if $\Delta_1$ and $\Delta_2$ are simple systems, then some Weyl-group element sends $\Delta_1$ to $\Delta_2$. Weyl-group elements preserve the root inner product, and hence preserve the Cartan matrix up to relabelling. Thus the Dynkin diagram is independent of both choices and is an invariant $D(\mathfrak{g})$ of the isomorphism class of $\mathfrak{g}$.[/guided]

[step:Identify the possible connected components] A semisimple Lie algebra decomposes as a direct sum of simple ideals, \begin{align*} \mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_r. \end{align*} For compatible Cartan subalgebras $\mathfrak{h}_i \subset \mathfrak{g}_i$, the subalgebra $\mathfrak{h} := \mathfrak{h}_1 \oplus \cdots \oplus \mathfrak{h}_r$ is a Cartan subalgebra of $\mathfrak{g}$ and \begin{align*} \Phi(\mathfrak{g},\mathfrak{h}) = \Phi(\mathfrak{g}_1,\mathfrak{h}_1) \sqcup \cdots \sqcup \Phi(\mathfrak{g}_r,\mathfrak{h}_r) \end{align*} with orthogonal components. Therefore $D(\mathfrak{g})$ is the disjoint union of the connected diagrams $D(\mathfrak{g}_i)$. The classification theorem for irreducible finite reduced crystallographic root systems says that the connected finite-type Dynkin diagrams are exactly \begin{align*} A_n,\ B_n,\ C_n,\ D_n,\ E_6,\ E_7,\ E_8,\ F_4,\ G_2, \end{align*} with the usual rank restrictions on the classical families. Hence every semisimple Lie algebra gives a finite disjoint union of diagrams from the stated list. [/step]

[step:Construct a Lie algebra from every allowed diagram]Let $D$ be a finite disjoint union of connected Dynkin diagrams from the stated list, and let $A_D$ be its finite-type generalized Cartan matrix. [Serre's presentation theorem](/theorems/4679) associates to $A_D$ a finite-dimensional semisimple Lie algebra $\mathfrak{g}(A_D)$ over $k$, generated by elements $e_i,f_i,h_i$ indexed by the vertices and subject to the Cartan and Serre relations determined by $A_D$. The theorem applies because $A_D$ is of finite type, equivalently its symmetrization is positive definite. The root system of $\mathfrak{g}(A_D)$ has simple-root Cartan matrix $A_D$, so its Dynkin diagram is $D$. Thus every finite disjoint union in the list is attained.[/step]

[guided]We now prove surjectivity of the classification map. Start with a finite disjoint union $D$ of diagrams from the list. The diagram determines a finite-type generalized Cartan matrix $A_D$: diagonal entries are $2$, off-diagonal entries are nonpositive integers, and the number and direction of edges encode the products $a_{ij}a_{ji}$ and relative root lengths. Because each connected component of $D$ is one of the finite Dynkin diagrams, the matrix $A_D$ is of finite type; equivalently, after symmetrization it defines a positive definite form. This verifies the finite-type hypothesis in Serre's presentation theorem. Serre's theorem constructs a Lie algebra $\mathfrak{g}(A_D)$ over $k$ with generators $e_i,f_i,h_i$ indexed by vertices and relations determined by $A_D$, and it asserts that in finite type this Lie algebra is finite-dimensional and semisimple. The same theorem identifies the simple roots of $\mathfrak{g}(A_D)$ with the vertices of $D$ and gives Cartan matrix $A_D$. Therefore the Dynkin diagram attached to $\mathfrak{g}(A_D)$ is exactly $D$. This proves that every diagram in the claimed list occurs.[/guided]

[step:Use uniqueness to obtain injectivity] Suppose $\mathfrak{g}$ and $\mathfrak{g}'$ are finite-dimensional semisimple Lie algebras over $k$ with the same Dynkin diagram. Choose Cartan subalgebras and simple systems so that their Cartan matrices are the same matrix $A$. The uniqueness part of Serre's theorem, equivalently the uniqueness theorem for finite-dimensional semisimple Lie algebras with a fixed finite-type root datum, gives Lie algebra isomorphisms \begin{align*} \mathfrak{g} \cong \mathfrak{g}(A) \cong \mathfrak{g}'. \end{align*} Thus the diagram invariant is injective on isomorphism classes. [/step]

[step:Conclude the classification] The previous steps define a map from isomorphism classes of finite-dimensional semisimple Lie algebras over $k$ to finite disjoint unions of Dynkin diagrams from the stated list. The construction from Serre's theorem proves that this map is surjective, and the uniqueness theorem proves that it is injective. Since direct sums of simple ideals correspond to orthogonal disjoint unions of irreducible root systems, and hence to disjoint unions of connected Dynkin diagrams, the map is compatible with decomposition into simple summands. Therefore finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic $0$ are classified, up to isomorphism, by finite disjoint unions of Dynkin diagrams of types $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$. [/step]