  We study the moduli stack of degree $0$ semistable $G$-bundles on an
irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected
reductive group. Our main result describes a partition of this stack indexed by
a certain family of connected reductive subgroups $H$ of $G$ (the
$E$-pseudo-Levi subgroups), where each stratum is computed in terms of
$H$-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 $E$ has a
single cusp (respectively, node), this gives a new proof of the
Jordan--Chevalley theorem for the Lie algebra $\mathfrak{g}$ (respectively,
group $G$).
  We also provide a Tannakian description of these moduli stacks and use it to
show that if $E$ is an ordinary elliptic curve, the collection of framed
unipotent bundles on $E$ is equivariantly isomorphic to the unipotent cone in
$G$. Finally, we classify the $E$-pseudo-Levi subgroups using the Borel--de
Siebenthal algorithm and compute some explicit examples.
