  We discuss the conditions under which the BRST operator of a $W$-string can
be written as the sum of two operators that are separately nilpotent and
anticommute with each other. We illustrate our results with the example of the
non-critical $W_3$-string. Furthermore, we apply our results to make a
conjecture about a relationship between the spectrum of a non-critical
$W_n$-string and a $W_{n-1}$-string.
