  The theory of nilpotent orbits of simple Lie algebras has seen tremendous
developments over the past decades. In this context an important role is played
by the component group of the stabilizer of a nilpotent element. In this work,
the aim is to show computational methods to obtain explicit generators of the
component group of the centralizer of a nilpotent element in a simple Lie
algebra over $\mathbb{C}$. In some cases such generators had already been
determined by impressive hand calculations but these often use the specific
form of the chosen representative and are thus not immediately applicable to
different representatives of the same orbit. It is then interesting to show how
to overcome this issue constructing specific algorithms: for the classical
types there is a straightforward method that directly translates well-known
theoretical constructions; for the exceptional types we devise a method using
the double centralizer of an $\mathfrak{sl}_2$-triple. In particular, it gives
an independent construction of the component group, that does not depend on the
prior knowledge of its isomorphism type. For our purposes, we needed to
construct many algorithms which have been mainly implemented in the
computational algebra system GAP.
