Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected. The Wikipedia page[1] mentions the issue briefly but I was curious

Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected. The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices). [1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#Surjectivity_of_the_exponential https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S... [2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...