We demonstrate that certain Virasoro characters (and their linear
combinations) in minimal and non-minimal conformal models which admit
factorized forms are manifestly related to the ADE series. This permits to
extract quasi-particle spectra of a Lie algebraic nature which resembles the
features of Toda field theory. These spectra possibly admit a construction in
terms of the $W_n$-generators. In the course of our analysis we establish
interrelations between the factorized characters related to the parafermionic
models, the compactified boson and the minimal models.