Prove that two Lie groups have homeomorphic universal covers if and only if their corresponding Lie algebra are isomorphic

Question

Two Lie groups $G_1, G_2$ have homeomorphic universal covers $\tilde{G_1}, \tilde{G_2}$ respectively if and only if the corresponding Lie algebras $\frak{g_1}, \frak{g_2}$ are isomorphic as Lie groups.

This is based off of the Lie algebra-Lie group correspondence but I am not sure how to really show this is true. A proof of one direction would be very helpful.

Answer 1

This is not true, any simply-connected $n$-dimensional nilpotent Lie group is homeomorphic to $R^n$, but its Lie algebra is not always commutative. Thus is not isomorphic to the Lie algebra of the $n$-dimensional simply connected commutative Lie group $R^n$.