Unitary units and skew elements in group algebras

 Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g−1,gG. Let Un(FG)={uFG|uu*=1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG−, the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG− is Lie nilpotent and char F≠2.