What made the Lie exponential click for me is the definition of exp as a limit: exp(x) = (I + x / n)^n when n goes to infinity. A tangent vector x at the ident

What made the Lie exponential click for me is the definition of exp as a limit: exp(x) = (I + x / n)^n when n goes to infinity. A tangent vector x at the identity is split in n chunks, each chunk is added to the identity to get a matrix (I + x / n). Each of these matrices gets increasingly closer to the group as n increases, and these n matrices are composed together to approximate exp(x), in a similar fashion to forward Euler integration with n iterations and integration step 1/n. The actual formula ends up being the usual power series and is useful for computations, but I find the limit above really captures the essence of how the exponential works.