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Examples of non-closed subgroups are plentiful; for example take ''G'' to be a 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.


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