  We show that the moduli space of semistable G-bundles on an elliptic curve
for a reductive group G is isomorphic to a power of the elliptic curve modulo a
certain Weyl group which depend on the topological type of the bundle. This
generalises a result of Laszlo to arbitrary connected components and recovers
the global description of the moduli space due to Friedman--Morgan--Witten and
Schweigert. The proof is entirely in the realm of algebraic geometry and works
in arbitrary characteristic.
