Proposition 6

Let G be a Lie group and H a subgroup of G. For H to be a Lie subgroup of G, it is necessary and sufficient that there exist a point h ∈ H and an open neighbourhood U of h in G such that H ∩ U is a submanifold of G.

The condition is obviously necessary. Suppose that it holds. For all h' ∈ H, the translation γ(h'h^{-1}) is an automorphism of the manifold G and maps the submanifold H ∩ U of U into the submanifold (h'h^{-1}H) ∩ (h'h^{-1}U) of h'h^{-1}U. As h'h^{-1}H = H and h'h^{-1}U is an open neighbourhood of h' in G, we see that every point of H has an open neighbourhood V such that V ∩ H is a submanifold of G. Hence H is a submanifold of G.

Let G be a Lie group and H a Lie subgroup of G. If L is a Lie subgroup of H, L is a Lie subgroup of G by Differentiable and Analytic Manifolds, R, 5.8.6. Let M be a Lie subgroup of G such that M ⊂ H. Then M is a Lie subgroup of H, for the canonical injection of M into H is obviously an immersion.

Let k be a non-discrete closed subfield of K. A Lie k-subgroup of G is a Lie subgroup of the underlying Lie k-group of G.