Examples of non-closed subgroups are plentiful; for example take G to be a torus of dimension 2 or greater, 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 φ : R → → G with i m ( φ φ ) = H . The closure of H will be a sub-torus in G .