  We discuss a consequence of Green and Tao's factorisation theorem for
polynomial orbits on nilmanifolds, adjusted to the requirements of certain
arithmetic applications. More precisely, we prove a generalisation of Theorem
16.4, Acta Arith. 154 (2012), 235-306, by slightly rearranging its proof. The
thus achieved strengthening of the result removes an oversight in the
above-cited paper which resulted from the previously too weak conclusion. Since
this type of result proved essential for further applications, we take the
opportunity to discuss it in more detail.
