  The index of a finite-dimensional Lie algebra $g$ is the minimum of
dimensions of stabilisers $g_\alpha$ of elements $\alpha\in g^*$. Let $g$ be a
reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$.
Elashvili has conjectured that the index of the centraliser $z(x)$ equals the
index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for
reductive Lie algebras of classical type.
  It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of
$z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent
element $x\in so_8$ such that the coadjoint action of $z(x)$ has no generic
stabiliser.
