Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy of $\mathrm{SL}(2,\mathbb{C})$ such that

$\bullet$ its action preserves a decomposition $\mathbb{C}^n = \mathbb{C}^2 \oplus \mathbb{C}^{n-2}$

$\bullet$ its action is the natural action of $\mathrm{SL}(2,\mathbb{C})$ on $\mathbb{C}^2$ and it acts trivially on $\mathbb{C}^{n-2}$.

My question is: must $\rho(G)$ be equal to the whole $\mathrm{SL}(n,\mathbb{C})$? or do you know any counterexamples?

Obviously, $\mathrm{SL}(n,\mathbb{C})$ contains $\mathrm{O}(n, \mathbb{C})$ the group of automorphisms of a non degenerate quadratic form and $\mathrm{Sp}(n, \mathbb{C})$ the symplectic group if $n$ is even. So for my question not to be trivial, I make the extra assumption that $\rho(G)$ is not contained in any conjugate of these two groups.

I understand that diging deep enough in the theory of representations should provide an answer to my question, but being no expert in Lie theory I was wondering if I was missing any simple argument or counter-example. Thanks for your attention.