We describe a method for associating to a Lie algebra $\mathfrak{g}$ over a ring $\mathbb{K}$ a sequence of groups $(G_{n}(\mathfrak{g}))_{n\in\mathbb{N}}$, which are {\it polynomial groups} in the sense of Definition \ref{polygroup}. Using a description of these groups by generators and relations, we prove the existence of an action of the symmetric group $\Sigma_{n}$ by automorphisms. The subgroup of fixed points under this action, denoted by $J_{n}(\mathfrak{g})$, is still a polynomial group and we can form the projective limit $J_{\infty}(\mathfrak{g})$ of the sequence $(J_{n}(\mathfrak{g}))_{n\in\mathbb{N}}$. The formal group $J_{\infty}(\mathfrak{g})$ associated in this way to the Lie algebra $\mathfrak{g}$ may be seen as a generalisation of the formal group associated to a Lie algebra over a field of characteristic zero by the Campbell-Haussdorf formula.
Construction of groups associated to Lie- and to Leibniz-algebras