Earlier, the author described up to conjugacy all pairs AB of nilpotent subgroups of a finite group G with socle L2(q) for which AsuperscriptBg1 for any element of G. A similar description was obtained by the author later for primary subgroups A and B of a finite group G with socle Ln(2m). In this paper, we describe up to conjugacy all pairs AB of nilpotent subgroups of a finite group G with simple socle from the “Atlas of Finite Groups” for which AsuperscriptBg1 for any element g of G. The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group G for any nilpotent subgroups N contains an element g such that N ∩ Ng=1.