with on the diagonal. Then the decomposition is given by where for the vector in with the standard (matrix) basis, meaning represents the basis vector in the -th row and -th column. This decomposition of has an associated root system: sl2(C) For example, in the decomposition is and the associated root system is sl3(C) In the decomposition is and the associated root system is given by Examples As noted in #Structure, semisimple Lie algebras over (or more generally an algebraically closed field of characteristic zero) are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams. Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are: , the special linear Lie algebra. , the odd-dimensional special orthogonal Lie algebra. , the symplectic Lie algebra. , the even-dimensional special orthogonal Lie algebra (). The restriction in the family is needed because is one-dimensional and commutative and therefore not semisimple. These Lie algebras are numbered so that n is the rank. Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank. For example and . These four families, together with five exceptions (E6, E7, E8, F4, and G2), are in fact the only simple Lie algebras over the complex numbers. Classification Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – An, Bn, Cn, and Dn – with five exceptions E6, E7, E8, F4, and G2. Simple Lie algebras are classified by the connected Dynkin diagrams, shown on the right, while semisimple Lie algebras correspond to not necessarily connected Dynkin diagrams, where each component of the diagram corresponds to a summand of the decomposition of the semisimple Lie algebra into simple Lie algebras. The classification proceeds by considering a Cartan subalgebra (see below) and its adjoint action on the Lie algebra. The root system of the action then both determines the original Lie algebra and must have a very constrained form, which can be classified by the Dynkin diagrams. See the section below describing Cartan subalgebras and root systems for more details. The classification is widely considered one of the most elegant results