  For complex parallelisable manifolds $\Gamma\backslash G$, with $G$ a
solvable or semisimple complex Lie group, the Fr\"olicher spectral sequence
degenerates at the second page. In the solvable case, the de-Rham cohomology
carries a pure Hodge structure. In contrast, in the semisimple case, purity
depends on the lattice, but there is always a direct summand of the de Rham
cohomology which does carry a pure Hodge structure and is independent of the
lattice.
