We describe enveloping algebras of finite-dimensional Lie algebras which are formal in the sense that their Hochschild complex as a differential graded Lie algebra is quasi-isomorphic to its Hochschild cohomology. For Abelian Lie algebras this is true thanks to the Kontsevich formality theorem. We are using his formality map twisted by the group-like element generated by the linear Poisson structure to simplify the problem, and then study examples. For instance, the universal enveloping algebras of the Lie algebras \(\mathfrak{gl}(n,\mathbb{K})\oplus\mathbb{K}^{n}\) are formal. We also recover our rigidity results for enveloping algebras from this new angle and present some explicit deformations of linear Poisson structure in low dimensions.