  We simplify and generalize an argument due to Bowcock and Watts showing that
one can associate a finite Lie algebra (the `classical vacuum preserving
algebra') containing the M\"obius $sl(2)$ subalgebra to any classical
$\W$-algebra. Our construction is based on a kinematical analysis of the
Poisson brackets of quasi-primary fields. In the case of the $\W_\S^\G$-algebra
constructed through the Drinfeld-Sokolov reduction based on an arbitrary
$sl(2)$ subalgebra $\S$ of a simple Lie algebra $\G$, we exhibit a natural
isomorphism between this finite Lie algebra and $\G$ whereby the M\"obius
$sl(2)$ is identified with $\S$.
