  We study the geometrical structure of the coadjoint orbits of an arbitrary
complex or real Lie algebra ${\mathfrak g}$ containing some ideal ${\mathfrak
n}$. It is shown that any coadjoint orbit in ${\mathfrak g}^*$ is a bundle with
the affine subspace of ${\mathfrak g}^*$ as its fibre. This fibre is an
isotropic submanifold of the orbit and is defined only by the coadjoint
representations of the Lie algebras ${\mathfrak g}$ and ${\mathfrak n}$ on the
dual space ${\mathfrak n}^*$. The use of this fact and an application of
methods of symplectic geometry give a new insight into the structure of
coadjoint orbits and allow us to generalize results derived earlier in the case
when ${\mathfrak g}$ is a split extension using the Abelian ideal ${\mathfrak
n}$ (a semidirect product). As applications, a new proof of the formula for the
index of Lie algebra and a necessary condition of integrality of a coadjoint
orbit are obtained.
