A note on Galois representations valued in reductive groups with open image

Let G be a split reductive group with dim⁡Z(G)≤1. We show that for any prime p that is large enough relative to G, there is a finitely ramified Galois representation ρ:ΓQ→G(Zp) with open image. We also show that for any given integer e, if the index of irregularity of p is at most e and if p is large enough relative to G and e, then there is a Galois representation ρ:ΓQ→G(Zp) ramified only at p with open image, generalizing a theorem of Ray [8]. The first type of Galois representation is constructed by lifting a suitable Galois representation into G(Fp) using a lifting theorem of Fakhruddin–Khare–Patrikis [4], and the second type of Galois representation is constructed using a variant of the argument in Ray's work [8].