While the Euclidean space R^{n} is a Lie group, much more important examples are groups of matrices, for instance the group SO(3) of all rotations in 3-dimensional space.
A vector field X on a Lie group G is said to be left invariant if it commutes with left translation: Define L_{g}[f](x)= f(gx) for any analytic function f : G -> R and all g, x in G. Then a vector field is left invariant if X L_{g} = L_{g} X for all g in G.
The set of all vector fields on an analytic manifold is a Lie algebra. The subalgebra of all left invariant vector fields is called the Lie algebra associated with G, and is usually denoted by a gothic g. This Lie algebra g is finite-dimensional (it has the same dimension as the manifold G) which makes it susceptible to classification attempts. By classifying g, one can also get a handle on the Lie group G.
Every element v of the tangent space T_{e} at the identity element e of G determines a unique left invariant vector field whose value at the element x of G will be denoted by xv; the vector space underlying g may therefore be identified with T_{e}. The vector v furthermore determines a function c : R -> G whose derivative everywhere is given by the corresponding left invariant vector field
The operation t, s -> tst^{-1}s^{-1} is called the commutator operator. Note that this operation sends (1,1) to 1, and so defines an operation on the tangent space at 1 by differentiation. The function it defines on the tangent space is bi-linear, and it turns out that it gives the tangent space a Lie algebra structure.
Sometimes, Lie groups are defined as topological manifolds with continuous group operations; the two definitions are equivalent. This is the content of Hilbert's fifth problem. The precise statement, proven by Gleason, Montgomery and Zippin in the 1950s, is as follows: If G is a topological manifold with continuous group operations, then there exists exactly one differentiable structure on G which turns it into a Lie group in our sense.