  We consider Drinfeld-Sokolov bihamiltonian structure associated to a
distinguished nilpotent elements of semisimple type and the space of common
equilibrium points defined by its leading term. On this space, we construct a
local bihamiltonian structure which form an exact Poisson pencil, defines an
algebraic classical $W$-algebra, admits a dispersionless limit, and its leading
term defines an algebraic Frobenius manifold. This leads to a uniform
construction of algebraic Frobenius manifolds corresponding to regular cuspidal
conjugacy classes in irreducible Weyl groups.
