  Let G be a simple algebraic group over an algebraically closed field with Lie
algebra g. Then the orbits of nilpotent elements of g under the adjoint action
of G have been classified. We describe a simple algorithm for finding a
representative of a nilpotent orbit. We use this to compute lists of
representatives of these orbits for the Lie algebras of exceptional type. Then
we give two applications. The first one concerns settling a conjecture by
Elashvili on the index of centralizers of nilpotent orbits, for the case where
the Lie algebra is of exceptional type. The second deals with minimal
dimensions of double centralizers.
