Representing Lie algebras using approximations with nilpotent ideals

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincar\'e-Birkhoff-Witt and the almost-algebraic hulls of Auslander and Brezin.