  The noncritical $D=4$ $W_3$ string is a model of $W_3$ gravity coupled to two
free scalar fields. In this paper we discuss its BRST quantization in direct
analogy with that of the $D=2$ (Virasoro) string. In particular, we calculate
the physical spectrum as a problem in BRST cohomology. The corresponding
operator cohomology forms a BV-algebra. We model this BV-algebra on that of the
polyderivations of a commutative ring on six variables with a quadratic
constraint, or, equivalently, on the BV-algebra of (polynomial) polyvector
fields on the base affine space of $SL(3,C)$. In this paper we attempt to
present a complete summary of the progress made in these studies. [...]
