  In recent work, the authors proved a general result on lifting
$G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to
G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$
a reductive group, to geometric $\ell$-adic representations. In this note we
take $G$ to be a classical group and construct many examples of $G$-irreducible
representations to which these new lifting methods apply, but to which the
lifting methods provided by potential automorphy theorems do not.
