Let $G$ be a reductive group over a number field $K$. Let $\Gamma\subset G(K)$ be a subgroup.

My extremely naive question is - When can you deduce that $\Gamma$ is Zariski-dense? I'm looking for criteria for Zariski-density, possibly making additional assumptions on $G,K,\Gamma$.

To my surprise I've found it to be very difficult to find information on this question. References would also be appreciated!

As a special case of this, let $R_K\subset K$ be the ring of integers. Fix an embedding of algebraic $K$-groups $i : G\hookrightarrow \text{GL}_{n,K}$, and define $G(R_K) := \text{GL}_n(R_K)\cap i(G(K))$. When is $G(R_K)$ Zariski-dense? Does it depend on the embedding $i$?

It might be appropriate to make this a community wiki