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: $E_6,E_7,E_8,F_4,G_2$.

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.

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 .