  In this paper we study the universal lifting spaces of local Galois
representations valued in arbitrary reductive group schemes when $\ell \neq p$.
In particular, under certain technical conditions applicable to any root datum
we construct a canonical smooth component in such spaces, generalizing the
minimally ramified deformation condition previously studied for classical
groups. Our methods involve extending the notion of isotypic decomposition for
a $\textrm{GL}_n$-valued representation to general reductive group schemes. To
deal with certain scheme-theoretic issues coming from this notion, we are led
to a detailed study of certain families of disconnected reductive groups, which
we call weakly reductive group schemes. Our work can be used to produce
geometric lifts for global Galois representations, and we illustrate this for
$\mathrm{G}_2$-valued representations.
