There is a three-way correspondence between:

• Real 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.