  Ado's Theorem had been extended to principal ideal domains independently by
Churkin and Weigel. They demonstrated that if $R$ is a principal ideal domain
of characteristic zero and $\mathfrak{L}$ is a Lie algebra over $R$ which is
also a free $R$-module of finite rank, then $\mathfrak{L}$ admits a finite
faithful Lie algebra representation over $R$.
  We present a quantitative proof of this result, providing explicit bounds on
the degree of the Lie algebra representations in terms of the rank of the free
module. To achieve it, we generalise an established embedding theorem for
complex Lie algebras: any Lie algebra as above embeds within a larger Lie
algebra that decomposes as the direct sum of its nilpotent radical and another
subalgebra.
