A remark on Ado's Theorem for principal ideal domains

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.