A good example of a Lie group is the group E3 of all isometries of euclidean space ℝ3. Euclidean geometry is the study of those properties of subsets of ℝ3 which are preserved when the subset is transformed by an element of E3, so to know what E3 is is the same thing as to know what is meant by Euclidean geometry. In general, Lie groups are the basic tools of geometry.

Besides being a group a crucial property of E3 is that it has a topology, i.e. it makes sense to say that one element is “near” another, or to speak of a “continuous path” in E3. Thus E3 consists of two connected components, one formed by the elements which preserve orientation and the other by those which reverse it, and there is no continuous path from one of the former to one of the latter.

A simpler example is the subgroup 03 of E3 consisting of isometries of ℝ3 which leave the origin fixed. This can, of course, be identified with the group of 3 × 3 real orthogonal matrices A. Again it consists of two connected components, the subgroup of matrices A with determinant +1, which is called SO3, and the coset of matrices with determinant -1. The group SO3 consists of all rotations about axes through the origin in ℝ3. A rotation is determined by its axis and the angle of rotation, which is taken between 0 and π.