In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartan's theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is a Lie subalgebra of the Lie algebra g of G.
Examples of non-closed subgroups are plentiful; for example take G to be a torus of dimension ≥ 2, and let H be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism φ : R → G with H as its image. The closure of H will be a sub-torus in G.
In terms of the exponential map of G, in general, only some of the Lie subalgebras of the Lie algebra g of G correspond to Lie subgroups H of G. There is no criterion solely based on the structure of g which determines which those are.