If is any subgroup of "G", not necessarily a closed subgroup, then is a Lie subalgebra of and, as a consequence of the Lie correspondence, the connected component of is the group generated by.
If is any subgroup of "G", not necessarily a closed subgroup, then is a Lie subalgebra of; here is the exponential map from the Lie algebra to the Lie group. As a consequence of the Lie correspondence, the connected component of is the closed subgroup of "G" generated by.