  Let $X$ be a smooth irreducible projective curve. Recently, Pauly and
Pe\'on-Nieto shows that a vector bundle over $X$ is very stable if and only if
the Hitchin map on the vector space of Higgs field on that vector bundle is
proper. In this notes, we generalize this result to principal $G-$bundles for
any semisimple linear algebraic group $G$. We also study the relation between
very stability and other stability conditions in the case of
$\text{SL}_2-$bundles.
