  We study the Belkale-Kumar family of cup products on the cohomology of a
generalized flag variety. We give an alternative construction of the family
using relative Lie algebra cohomology, and in particular, identify the
Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every
value of the parameter. As a consequence, we extend a fundamental disjointness
result of Kostant to a family of Lie algebras. In an appendix, written jointly
with Edward Richmond, we extend a Levi movability result of Belkale and Kumar
to arbitrary parameters.
