  Let $G$ be a group and let $F$ be a field of characteristic different from 2.
Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the
set of symmetric units, under an oriented classical involution of the group
algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index
of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.
