  In the present paper we study twisted foldings of root systems which
generalize usual involutive foldings corresponding to automorphisms of Dynkin
diagrams. Our motivating example is the Lusztig projection of the root system
of type $E_8$ onto the subring of icosians of the quaternion algebra which
gives the root system of type $H_4$.
  Using moment graph techniques for any such folding we construct a map at the
equivariant cohomology level. We show that this map commutes with
characteristic classes and Borel maps. We also introduce and study its
restrictions to the usual cohomology of projective homogeneous varieties, to
group cohomology and to their virtual analogues for finite reflection groups.
