  We illustrate the Lie theoretic capabilities of the computational algebra
system GAP4 by reporting on results on nilpotent orbits of simple Lie algebras
that have been obtained using computations in that system. Concerning reachable
elements in simple Lie algebras we show by computational means that the simple
Lie algebras of exceptional type have the Panyushev property. We
computationally prove two propositions on the dimension of the abelianization
of the centralizer of a nilpotent element in simple Lie algebras of exceptional
type. Finally we obtain the closure ordering of the orbits in the null cone of
the spinor representation of the group Spin_{13}(C). All input and output of
the relevant GAP sessions is given.
