  For any simple algebraic group $G$ of exceptional type, we construct
geometric $\ell$-adic Galois representations with algebraic monodromy group
equal to $G$, in particular producing the first such examples in types
$\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive
groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois
representations to geometric $\ell$-adic representations.
