  We prove that the classical $W$-algebra associated to a nilpotent orbit in a
simple Lie-algebra can be constructed by preforming bihamiltonian,
Drinfeld-Sokolov or Dirac reductions.
  We conclude that the classical $W$-algebra depends only on the nilpotent
orbit but not on the choice of a good grading or an isotropic subspace.
  In addition, using this result we prove again that the transverse Poisson
structure to a nilpotent orbit is polynomial and we better clarify the relation
between classical and finite $W$-algebras.
