Realisations of W3 symmetry

We perform a systematic investigation of free-scalar realisations of the Zamolodchikov W3 algebra in which the operator product of two spin-three generators contains a non-zero operator of spin four which has vanishing norm. This generalises earlier work where such an operator was required to be absent. By allowing this spin-four null operator we obtain several realisations of the W3 algebra both in terms of two scalars as well as in terms of an arbitrary number n of free scalars. Our analysis is complete for the case of two-scalar realisations.