Faithful representations of Lie algebras and Homogeneous Spaces

The faithful representations of real Lie algebras g with dimensions $n\le 4$ acting on the dual space of the universal covering algebra of an ideal s of g, are calculated. With the help of left and right translations of the Lie group, we define the homogeneous manifolds of dimension n and we calculate the Killing fields (generators of left translations), the invariant fields (generators of right translations) and the left invariant 1-forms.