Pink-type results for general subgroup of ${\mathrm{SL}}_2({\mathbb{Z}}_{\mathrm{\ell }}{)}^n$

We study open subgroups $G$ of ${\mathrm{SL}}_2({\mathbb{Z}}_{\mathrm{\ell }}{)}^n$ in terms of some associated Lie algebras without assuming that $G$ is a pro-$\mathrm{\ell }$ group, thereby extending a theorem of Pink. The result has applications to the study of families of Galois representations.