An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is non-abelian and simple. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.
Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.
While the notion of a simple Lie group is satisfying from the axiomatic perspective, in applications of Lie theory, such as the theory of Riemannian symmetric spaces, somewhat more general notions of semisimple and reductive Lie groups proved to be even more useful. In particular, every connected compact Lie group is reductive, and the study of representations of general reductive groups is a major branch of representation theory.
The most common definition is the one above: simple Lie groups have to be connected, they are allowed to have non-trivial centers (possibly infinite), they need not be representable by finite matrices, and they must be non-abelian.
Such groups are classified using the prior classification of the complex simple Lie algebras: for which see the page on root systems. It is shown that a simple Lie group has a simple Lie algebra that will occur on the list given there, once it is complexified (that is, made into a complex vector space rather than a real one). This reduces the classification to two further matters.
The groups SO(p,q,R) and SO(p+q,R), for example, give rise to different real Lie algebras, but having the same Dynkin diagram. In general there may be different real forms of the same complex Lie algebra.
Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Élie Cartan.
For an example, take the special orthogonal groups in even dimension. With the non-identity matrix −I in the center, these aren't actually simple groups; and having a two-fold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above.
According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes:
Dr corresponds to the special orthogonal group, SO(2r). Note that SO(4) is not a simple group, though. The Dynkin diagram has two nodes that are not connected. There is a surjective homomorphism from SO(3)* × SO(3)* to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Therefore the simple groups here start with D3, which as a diagram straightens out to A3. With D4 there is an 'exotic' symmetry of the diagram, corresponding to so-called triality.
See also E7½