Let's say I want to prove that a closed subgroup of GL(n,R) or GL(n,C) is a Lie group, with an atlas given by exponential of matrices (restricted to an appropriate subalgebra of gl(n)), without using any manifold or Lie theory. Can you provide the necessary argument? Maybe it's trivial, but I can't see it at the moment.
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Well, the crux of the matter is to cook up the linear subspace of gl(n) that will eventually be the subalgebra. If H is the subgroup: set h={X\in gl(n): exp(tX) \in H for all t \in\R}. You need to show two not completely obvious things:
Adams has ingenious short arguments for these on pages 1719 of his Lectures on Lie groups which require nothing but a little real analysis. I remark that all this uses no manifold or Lie theory beyond the necessary definitions and the argument works completely without change for closed subgroups of any Lie group. 

