  We show that a torsion-free nilpotent loop (that is, a loop nilpotent with
respect to the dimension filtration) has a torsion-free nilpotent left
multiplication group of, at most, the same class. We also prove that a free
loop is residually torsion-free nilpotent and that the same holds for any free
commutative loop. Although this last result is much stronger than the usual
residual nilpotency of the free loop proved by Higman, it is proved,
essentially, by the same method.
