  Let $k$ be a field of characteristic different from $2$ and let $G$ be a
nonabelian residually torsion-free nilpotent group. It is known that $G$ is an
orderable group. Let $k(G)$ denote the subdivision ring of the Malcev-Neumann
series ring generated by the group algebra of $G$ over $k$. If $\ast$ is an
involution on $G$, then it extends to a unique $k$-involution on $k(G)$. We
show that $k(G)$ contains pairs of symmetric elements with respect to $\ast$
which generate a free group inside the multiplicative group of $k(G)$. Free
unitary pairs also exist if $G$ is torsion-free nilpotent. Finally, we consider
the general case of a division ring $D$, with a $k$-involution $\ast$,
containing a normal subgroup $N$ in its multiplicative group, such that $G
\subseteq N$, with $G$ a nilpotent-by-finite torsion-free subgroup that is not
abelian-by-finite, satisfying $G^{*}=G$ and $N^{*}=N$. We prove that $N$
contains a free symmetric pair.
