  We study the cohomology of the critical $W_4$ string using the $W_4$ 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-four constraint only,
and it contains operators of the $c={4\over5}$ $W_3$ minimal model. In the next
step, where the spin-three constraint is added, these operators get dressed to
operators of the $c={7\over10}$ Virasoro minimal model. Finally, the Virasoro
constraint is added to obtain the cohomology of the critical $W_4$ string. We
describe the structure of the complete cohomology and compare with other
results.
