For a simple Lie algebra $\mathfrak{g}$ who contains a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, I’m trying to show that a nontrivial irreducible representation $\pi:\mathfrak{g}\to\mathrm{End}(V)$ is faithful.
I tried to apply the fact that $\pi$ is isomorphic to $\vee^k\mathbb{C}^2$ for any irreducible complex representation of $\mathfrak{sl}(2,\mathbb{R})$ but failed. Can someone please give me some hints to guide me?
The kernel of $\pi$ is an ideal of $\mathfrak g$. Since $\mathfrak g$ is simple, the ideal $\ker\pi$ is either $0$ or $\mathfrak g$. Since $\pi$ is non-trivial, $\ker\pi\ne\mathfrak g$, hence $\ker\pi=0$. This means $\pi$ is faithful.