  Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the
complexification of its maximal compact subgroup $H\subset G$. We consider
classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus
$g\geq2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This
allows us to find obstructions to a deformation retract from the moduli space
of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles
over $X$, in contrast with the situation when $g=1$, and to show reducibility
of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$
complex.
