  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.
