  We clarify the notion of the DS --- generalized Drinfeld-Sokolov ---
reduction approach to classical ${\cal W}$-algebras. We first strengthen an
earlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\cal
G}$ can be associated to every DS reduction. We then use the fact that a
$\W$-algebra must have a quasi-primary basis to derive severe restrictions on
the possible reductions corresponding to a given $sl(2)$ embedding. In the
known DS reductions found to date, for which the $\W$-algebras are denoted by
${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, the
quasi-primary basis corresponds to the highest weights of the $sl(2)$. Here we
find some examples of noncanonical DS reductions leading to $\W$-algebras which
are direct products of ${\cal W}_{\cal S}^{\cal G}$-algebras and `free field'
algebras with conformal weights $\Delta \in \{0, {1\over 2}, 1\}$. We also show
that if the conformal weights of the generators of a ${\cal W}$-algebra
obtained from DS reduction are nonnegative $\Delta \geq 0$ (which is
