Let $k$ be a field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$. $\mathfrak{g}$ corresponds to a formal group scheme $\mathcal{G} = \text{Spf} (U(\mathfrak{g})^{*})$. In fact, there is an equivalence of categories between finite-dimensional Lie algebras and infinitesimal formal group schemes over $k$ with finite-dimensional tangent space. This is elaborated on here by Akhil Mathew.

There should be an exponential map $\text{exp} : \mathfrak{g} \rightarrow \mathcal{G}$. This is mentioned at the end of the blog post above, but I can't figure out how to construct it explicitly.

Any help on this would be much appreciated.