  For any Lie algebra L over a field, its universal enveloping algebra U(L) can
be embedded in a division ring D(L) constructed by Lichtman. If U(L) is an Ore
domain, D(L) coincides with its ring of fractions. It is well known that the
principal involution of L, $x\mapsto -x$, can be extended to an involution of
U(L), and Cimpric has proved that this involution can be extended to one on
D(L). For a large class of noncommutative Lie algebras L over a field of
characteristic zero, we show that D(L) contains noncommutative free algebras
generated by symmetric elements with respect to (the extension of) the
principal involution. This class contains all noncommutative Lie algebras such
that U(L) is an Ore domain.
