Non-Trivial Non-Canonical W-Algebras from Kac-Moody Reductions
  By reducing a split $G_2$ Kac-Moody algebra by a non-maximal set of first-class constraints we produce W-algebras which (i) contain fields of negative conformal spin and (ii) are not trivial extensions of canonical W-algebras. 
G.A.T.F.da Costa and L.O'Raifeartaigh