Example

Let G be a Lie group, g its Lie algebra and H a Lie subgroup of G. Every left invariant vector field on H defines canonically a left invariant vector field on G, whence there is a canonical injection of the Lie algebra h of H into g; h is identified with a Lie subalgebra of g under this injection. If H is normal in G, the canonical image of h in g is an ideal of g.

An ideal of g is a submodule of g which is stable under the inner derivations of g.