  An important step in standard renormalisation arguments involve convolution
against a standard mollifier. As pointed out in (Punshon-Smith--Smith 2018),
this generates second order commutator terms in equations with gradient-type
noise. These are commutators similar to commutators in the well-known
``folklore lemma" of Di Perna--Lions (Di Perna--Lions 1989, Lemma II.1), but
not covered by standard renormalisation theory. In this note we establish the
vanishing of these commutators for gradient-type noises on $\mathbb{T}^d$ not
necessarily possessing divergence-free structure.
