  We study the unitarizability of premodular categories constructed from
representations of quantum group at roots of unity. We introduce
\emph{Grothendieck unitarizability} as a natural generalization of
unitarizability to any class of premodular categories with a common
Grothendieck semiring. We obtain new results for quantum groups of Lie types
$F_4$ and $G_2$, and improve the known results for Lie types $B$ and $C$.
