The Lie n-Engel property in group rings

Let FG be the group ring of a group G over a field F whose characteristic is p not equal 2. Let * denote the involution on FG which sends each group element to its inverse. Let (FG)(+) and (FG)(-) denote, respectively, the sets of symmetric and skew elements with respect to *. The conditions under which the group ring is Lie n-Engel for some n are known. We show that if either (FG)(+) or (FG)(-) is Lie n-Engel, and G is devoid of 2-elements, then FG is Lie m-Engel for some m. Furthermore, we completely classify the remaining groups for which (FG)(+) is Lie n-Engel.