Elements generating a free subgroup of rank three in the multiplicative group of a division ring

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.