BRST cohomology of the critical W string

We study the cohomology of the critical W4 string using the W4 BRST charge in a special basis in which it contains three separately nilpotent BRST charges. This allows us to obtain the physical operators in three steps. In the first step we obtain the cohomology associated to a spin-4 constraint only, and it contains operators of the c = 4/5 W3 minimal model. In the next step, where the spin-3 constraint is added, these operators get dressed to operators of the c = 7/10 Virasoro minimal model. Finally, the Virasoro constraint is added to obtain the cohomology of the critical W4 string. We describe the structure of the complete cohomology and compare with other results.