In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential fields extending a (...) model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations. (shrink)