On free subgroups of units of rings

We prove that if $a^{2}=b^{2}=0$ for elements $a,b$ of a ring $R$ of characteristic zero and $ab$ is not nilpotent, then there exists $m\in {\mathbf N}$ such that the group generated by $1+ma$ and $1+mb$ is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.