  In this work, we prove that, under a topological condition, the cohomology
associated with left-invariant elliptic structures on compact semisimple Lie
groups can be computed using only left-invariant forms. This reduces the
analytical problem to a purely algebraic one, while also providing a
generalization of the classic works of Chevalley and Eilenberg [CE48] on the de
Rham cohomology of compact Lie groups and of Pittie [Pit88] on the Dolbeault
cohomology of compact semisimple Lie groups to the context of elliptic
structures. We use spectral sequences as our primary tool, which facilitates
the construction of an isomorphism between the left-invariant differential
complex and the usual differential complex.
