When we discussed the Lie algebra of a Lie group we discussed “left-invariant” vector fields. More generally than this if is a diffeomorphism we say that a vector field is “-invariant” if it is -related to itself. That is, a vector field on a Lie group is left-invariant if it is -invariant for all .
Now we want a characterization of -invariance in terms of the flow of . I say that is -invariant if and only if for all . That is, the flow should commute with .
We’ll show this by showing that the vector field has flow . Then if is -related to itself we know that , and so by uniqueness we conclude that the flows and are equal, as asserted.
So, what makes the flow of ? First of all, we have to check the initial condition that , which is perfectly straightforward to check:
More involved is the differential condition. It will help if we rewrite a bit as a function of both and :
Now we can start on the differential condition:
And thus is indeed the flow of .