  In the paper "The Second cohomology of nilpotent orbits in classical Lie
algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P.
Chatterjee, and C. Maity, computable explicit descriptions of the second and
first cohomology groups of a general homogeneous space of a Lie group are
given, extending an earlier result in "On the exactness of Kostant-Kirillov
form and the second cohomology of nilpotent orbits, Internat. J. Math. 23
(2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. They turned out to be
crucial in our computations related to the cohomology of nilpotent orbits, as
done in the above two papers. Here we describe the third and fourth
cohomologies of a general homogeneous space along the line mentioned above. We
also draw various consequences of our main results in numerous special
settings. In particular, we obtain an interesting invariant by showing that for
a large class of homogeneous spaces, the difference between the third and
fourth Betti numbers coincides with the difference in the numbers of simple
factors of the ambient group and the associated closed subgroup.
