  Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and
let $\mu \in \mathfrak{g}^{*}$. In the first part of the paper, the relation is
discussed between the derived algebra of the stabilizer of $\mu$ and the set of
coadjoint orbits which have the same dimension as the orbit of $\mu$. In the
second part, semisimple Lie algebras are considered, and the relation is
discussed between the derived algebra of a centralizer and sheets.
