The cohomology groups of real toric varieties associated to Weyl chambers of type C and D

Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated to the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated to the Weyl chambers of type $C_n$ and $D_n$, completing the computation for all classical types.