Sections of Submonoids of Nilpotent Groups

We show that every product of f. g. \ submonoids of a group G is a section of a f. g. \ submonoid of GH₅ (Z), where H₅ (Z) is a Heisenberg group. This gives us a converse of a reduction of Bodart, and a new simple proof of the existence of a submonoid of a nilpotent group of class 2 with undecidable membership problem.