  The nilpotent cone of a reductive Lie algebra has a desingularization given
by thecotangent bundle of the flag variety. Analogously, the nullcone of a
cartesianpower of the algebra has a desingularization given by a vector bundle
over theflag variety. As for the nullcone, the subvariety of elements whose
componentsare in a same Borel subalgebra, has a desingularization given by a
vector bundle overthe flag variety. In this note, some properties of these
varieties are given. Forthe study of the commuting variety, the analogous
variety to the flag variety isthe closure in the Grassmannian of the set of
Cartan subalgebras. So someproperties of this variety are given. In particular,
it is smooth in codimension $1$.We introduce the generalized isospectral
commuting varieties and give some properties.Furthermore, desingularizations of
these varieties are given by fiber bundles over adesingularization of the
closure in the grassmannian of the set of Cartan subalgebrascontained in a
given Borel subalgebra.
