  We show that if a H\"{o}lder continuous linear cocycle over a hyperbolic
system is measurably conjugate to a cocycle taking values in a unipotent group,
then the cocycle is H\"older continuously conjugate to a cocycle taking values
in a unipotent group. More generally, we introduce some natural classes of
matrices contained in ${\rm GL}(d,\mathbb{R})$, which we call Zimmer blocks.
Examples of Zimmer blocks are unipotent and compact subgroups. We show that the
same conclusion holds for Zimmer blocks.
