Let G be a connected Lie group, LG its loop group, and π : PG → G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A = T (PG)/LG of this bundle. Given a symmetric bilinear form on \({\mathfrak{g}}\) and the corresponding central extension of \({L\mathfrak{g}}\) , we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form \({\eta \in \Omega^3(G)}\) arises as an obstruction. This involves the construction of a 2-form \({\varpi \in \Omega^{2}({\rm PG})^{\rm LG}= \Gamma(\wedge^2 A^*)}\) with \({{\rm d}\varpi=\pi^*\eta}\) . In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the form η, and for their G-equivariant extensions.