  Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on
compact Lie groups, we prove that, under certain algebraic and topological
conditions, the cohomology associated to left-invariant elliptic, and even
hypocomplex, involutive structures on compact Lie groups can be computed by
using only Lie algebras, thus reducing the analytical problem to a purely
algebraic one. The main tool is the Leray spectral sequence that connects the
result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault
cohomology of a homogeneous manifold.
