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We clarify the notion of the DS --- generalized Drinfeld-Sokolov ---reduction approach to classical ${\cal W}$-algebras. We first strengthen anearlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\calG}$ 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 onthe possible reductions corresponding to a given $sl(2)$ embedding. In theknown DS reductions found to date, for which the $\W$-algebras are denoted by${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, thequasi-primary basis corresponds to the highest weights of the $sl(2)$. Here wefind some examples of noncanonical DS reductions leading to $\W$-algebras whichare 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 showthat if the conformal weights of the generators of a ${\cal W}$-algebraobtained from DS reduction are nonnegative $\Delta \geq 0$ (which isComment: 48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-0

On the completeness of the set of classicalW-algebras obtained from DS reductions-algebras obtained from DS reductions