Nilpotency of groups of fibre homotopy equivalences in the rational category

Given F↪E→B a fibration we consider the group E♯(E,B) of self fibre homotopy equivalences which induce the identity in the homotopy groups of the fibre. Associated to a cofibration A↪X→X/A its Eckmann–Hilton dual its denoted by E∗(X,A). In this note we prove that, in the rational category and under suitable conditions, these groups are nilpotent.