Suppose we have two square complex matrices $X,Y $ in the lie algebra $\mathcal{G}$ of matrix Lie group $\mathbb G$ such that $e^X = e^Y$. Then $e^{tX}$ and $e^{tY}$ define the same one-parameter subgroup of $\mathbb G$, and $$X = \lim_{t\to 0} \dfrac{e^{tX}-I}{t} \stackrel{?}{=} \lim_{t\to 0} \dfrac{e^{tY}-I}{t} = Y$$
Presumably it is the middle equality which is incorrect. But I think I've seen this method used to prove that the exponential map for the Heisenberg group is injective. Is there some property of the Heisenberg group that makes this equality hold?