The author studies rings of constants for restricted differential Lie algebras acting on prime rings and having nontrivial quasi-Frobenius inner part. It is shown that, under this restriction, Shirshov's local finiteness correspondence exists between a given prime ring \(R\) and the ring of constants \(R^L\). The article under review is a continuation of the article by \textit{V.~K.~Kharchenko, J.~Keller}, and \textit{S.~Rodrigues-Romo} [Isr. J. Math. 96, Pt. B, 357-377 (1996; Zbl 0870.16021)]. Recall that a subring \(S\) of \(R\) is called a Shirshov right finite ring over a subring \(D\subseteq R\) if, for some elements \(v_1,\ldots,v_n\), the inclusion \(S\subseteq v_1D+\cdots+v_nD\) holds. A two-sided ideal \(A\) is said to be a Shirshov locally finite ideal over \(D\subseteq R\) if each finitely generated right ideal of \(R\) contained in \(A\) is a Shirshov finite ideal over \(D\). It is shown that the ring \(R\) has a nonzero ideal \(I\) such that each finitely generated right ideal contained in \(I\) is a submodule of a finitely generated right \(R^L\)-submodule of the module \(R\). In addition, the author studies the structure of \((R,R^L)\)-submodules contained in the Martindale left ring of quotients \(R_{\mathcal F}\) and proves the equalities \((R^L)_{\mathcal F}=(R_{\mathcal F})^L\) and \(Q(R^L)=Q(R)^L\) for the Martindale left and symmetric rings of quotients.