  We apply the filtered and graded methods developed in earlier works to find
(noncommutative) free group algebras in division rings.
  If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping
algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_L$ that contains
$U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_L$
generated by $U(L)$.
  Let $k$ be a field of characteristic zero and $L$ be a nonabelian Lie
$k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain,
we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras.
In those same cases, if $L$ is equipped with an involution, we are able to
prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated
by symmetric elements in most cases.
  Let $G$ be a nonabelian residually torsion-free nilpotent group and $k(G)$ be
the division subring of the Malcev-Neumann series ring generated by the group
algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$
contains a (noncommutative) free group algebra generated by symmetric elements.
