Classification of complex simple Lie algebras

Theorem (classification). Every finite-dimensional complex simple Lie algebra is isomorphic to exactly one of the Lie algebras in the following families:

Type $A_n$ ( $n\ge 1$ ): $\mathfrak{sl}_{n+1}(\mathbb{C})$ ,
Type $B_n$ ( $n\ge 2$ ): $\mathfrak{so}_{2n+1}(\mathbb{C})$ ,
Type $C_n$ ( $n\ge 3$ ): $\mathfrak{sp}_{2n}(\mathbb{C})$ ,
Type $D_n$ ( $n\ge 4$ ): $\mathfrak{so}_{2n}(\mathbb{C})$ ,
Exceptional types: $E_6,E_7,E_8,F_4,G_2$ .

Equivalent characterizations

Equivalently, complex simple Lie algebras are classified by connected Dynkin diagrams, or by indecomposable Cartan matrices satisfying the Cartan axioms.

Semisimple corollary. Every complex semisimple Lie algebra is a direct sum of simple ideals, so its isomorphism type is determined by a (finite) multiset of Dynkin diagram types.

Remarks

Context. The classification proceeds by choosing a Cartan subalgebra $\mathfrak{h}$ , analyzing the associated root system in $\mathfrak{h}^*$ , and encoding the relative geometry of simple roots in the Dynkin diagram. The root-system combinatorics precisely controls the Lie bracket via the root space decomposition.