@MikeMiller ok, thx. I have a different dumb question: I'm looking at a proof that for a connected, simply-connected Lie group $G$ with nilpotent Lie algebra $\mathfrak{g}$, we can equip $\mathfrak{g}$ with some product such that $G \cong \mathfrak{g}$. for the last step, they make use of a proposition that if two conn., simply-conn. Lie groups have isomorphic Lie algebras, they are isomorphic.

in this proposition, I assume I need a Lie algebra homomorphism and not just a vector space isomorphism, correct? (the latter would be easy to obtain since the exponential map is a diffeo in a nbhd …