In Link, Ribet gives a proof of Serre's open image theorem for elliptic curves using results from Faltings' proof of the Mordell conjecture. His argument is also repeated at https://mathoverflow.net/questions/108169/modern-proof-of-serres-open-image-theorem.
Unfortunately, I don't understand how he uses Faltings' Theorem to arrive at the punchline.
What does it mean for $\rho_{\ell, E}$ to be semisimple? I thought a semisimple representation was a direct sum of irreducibles, but Shafarevich's Theorem already guarantees that $V_{\ell}$ is irreducible.
What is the significant of the fact that $\text{End}_{\mathfrak{g}_{\ell}}(V_{\ell}) \simeq \mathbb{Q}_{\ell}$ (in the case of a non-CM elliptic curve)?
To summarize, how do these two results rule out the case of a non-split Cartan?