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In this paper we prove that for any connected reductive algebraic group G anda large enough prime $l$, there are continuous homomorphisms$$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ withZariski-dense image, in particular we produce the first such examples for$SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ and $E_7^{sc}$. To do this, we start with amod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ related tothe Weyl group of $G$ and use a variation of Stefan Patrikis' generalization ofa method of Ravi Ramakrishna to deform it to characteristic zero.

Algebraic monodromy groups of $l$-adic representations of $\mathrm{Gal}({\overline{\mathbb Q}}/{\mathbb Q})$