  The moduli space for a flat G-bundle over the two-torus is completely
determined by its holonomy representation. When G is compact, connected, and
simply connected, we show that the moduli space is homeomorphic to a product of
two tori mod the action of the Weyl group, or equivalently to the conjugacy
classes of commuting pairs of elements in G. Since the component group for a
non-simply connected group is given by some finite dimensional subgroup in the
centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin
diagram to prove properties of centralizers of pairs of elements in G, followed
by some explicit examples. We conclude by showing that as a result of a
compact, connected, simply connected Lie group G having a finite number of
subgroups, each conjugate to the centralizer of any element in G, that there is
a uniform bound on an irredundant chain of commuting elements.
