We study the moduli stack of degree
0
0
semistable
G
G
-bundles on an irreducible curve
E
E
of arithmetic genus
1
1
, where
G
G
is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups
H
H
of
G
G
(the
E
E
-pseudo-Levi subgroups), where each stratum is computed in terms of
H
H
-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where
E
E
has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra
g
\mathfrak {g}
(respectively, algebraic group
G
G
).
We also provide a Tannakian description of these moduli stacks and use it to show that if
E
E
is not a supersingular elliptic curve, the moduli of framed unipotent bundles on
E
E
are equivariantly isomorphic to the unipotent cone in
G
G
. Finally, we classify the
E
E
-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.