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Let F be a field of characteristic different from 2 and G a group. Under the classical involution on the group ring FG , we show that if FG is modular, then the group of unitary units of FG is nilpotent if and only if the entire unit group is nilpotent. We also demonstrate that this does not necessarily hold if FG is not modular, but it is still true if F is algebraically closed.

Let $KG$ be a non-commutative Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. A. Shalev [\emph{The derived length of Lie soluble group rings I}, J. Pure and Appl. Algebra {\bf{78}} (1992), 291-300] proved that $dl_L(KG)\geq \lceil \log_{2}(p+1)\rceil $ and posed the question of characterizing group algebras for which this lower bound is achieved. In this note the solution to this question is given.