TITLE: Is it possible to construct a formal group law from a Lie group without choosing coordinates?
QUESTION [11 upvotes]: There is a three-way correspondence between:

Real (connected and simply connected) Lie groups of dimension $n$;
$\mathbb R$-Lie algebras of dimension $n$;
Formal group laws in $n$ variables over the reals.

To get from the first to the second, one only really needs the fact that manifolds have tangent spaces which are vector bundles, and that the tangent space functor is strong (so that one can talk about left-invariant vector fields).
To get from the second to the third, one uses the Baker--Campbell--Hausdorff formula, which in the realm of formal power series is pure abstract algebra. 
However, the only way I know of getting from the first to the third directly involves choosing analytic coordinates around the identity and Taylor expanding the Lie group multiplication.
My question is: can one give a direct geometric construction of a formal group law out of a Lie group without choosing coordinates? The sort of thing I had in mind might, for example, involve using jet bundles.

REPLY [9 votes]: Sure: if $G$ is the given analytic group then consider the functor $F$ on finite local $\mathbf{R}$-algebras with residue field $\mathbf{R}$ given by defining $F(A)$ to be the set of maps of real-analytic spaces ${\rm{Sp}}(A) \rightarrow G$ "based" at the identity.  This is a group-valued functor, and it is visibly pro-represented by the completion $O_{G,e}^{\wedge}$ of the noetherian analytic local ring $O_{G,e}$ of $G$ at the identity $e$.  The group law on the functor thereby defines a group-object structure on the formal scheme Spf($O_{G,e}^{\wedge}$), or more specifically defines a formal Hopf algebra structure on $O_{G,e}^{\wedge}$ (i.e., a map $m^{\ast}:O_{G,e}^{\wedge} \rightarrow O_{G,e}^{\wedge} \widehat{\otimes} O_{G,e}^{\wedge}$ satisfying the usual conditions). 
Voila, that's it.  The same procedure works for analytic groups over non-archimedean fields in char. $p > 0$ (such groups might not be smooth, so "local coordinates" might not even exist, yet the method still works), as well as for group schemes of finite type over any field, and so on.  No need for Lie algebras or a BCH formula. The notion of "formal group" makes good sense in terms of complete local noetherian rings even without any notion of formal smoothness (so no "local coordinates").