On ordering the groups with nilpotent commutant

We prove that every group with nilpotent commutant, having an abelian normal subgroup such that the factor by this subgroup is nilpotent, is preorderable if and only if the group is $\Gamma$-torsion-free. An example is exhibited of a nonorderable $\Gamma$-torsion-free group with two-step nilpotent radical. This example demonstrates that for the variety of groups with nilpotent commutant the absence of $\Gamma$-torsion in a group is not a sufficient condition for orderability.