  We investigate infinitesimal properties of sets of ordered $n$-uples of
idempotents in a symmetric Banach $*$-algebra. These sets are called flag
manifolds and carry several interesting bundles that hold an important role in
some areas of operator theory. In this direction, we introduce and study
Stiefel bundles on flag manifolds, which are extensions of the well known
Stiefel bundles on Grassmannians. The main ingredient of our investigation is
the notion of connection on an infinite-dimensional bundle, and we survey some
equivalent ocurrences of such a notion in the literature.
