On a connection used in deformation quantization

Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.