Lie properties of symmetric elements in group rings II

Let FF be a field of characteristic different from 2, and GG a group with involution ∗∗. Write (FG)+(FG)+ for the set of elements in the group ring FGFG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if GG has no 2-elements, and (FG)+(FG)+ is Lie nilpotent (resp. Lie nn-Engel), then FGFG is Lie nilpotent (resp. Lie mm-Engel, for some mm). Here, we classify the groups containing 2-elements such that (FG)+(FG)+ is Lie nilpotent or Lie nn-Engel.