The crucial point in the proof is that the absolute Galois group $G_K$ acts irreducibly on $V_{\ell}$, which is based on a nontrivial Shafarevich finiteness theorem (Sect. 1.4). Applying this result to all finite algebraic extensions of $K$, one gets that the $\ell$-adic Lie algebra $g_{\ell}$ of the image acts irreducibly (and faithfully) on $V_{\ell}$.

So, $g_{\ell}$ is an irreducible Lie subalgebra of $\mathrm{End}_{\mathbb{Q}_{\ell}}(V_{\ell})$, whose centralizer consists of scalars $\mathbb{Q}_{\ell}\mathrm{Id}$, i.e., the natural faithful 2-dimensional representation of $g_{\ell}$ in $V_{\ell}$ is absolutely irreducible. Hence, $g_{\ell}$ is reductive, i.e., splits into a direct sum $$g_{\ell}=g_{\ell}^{0}\oplus c_{\ell}$$ of a semisimple Lie algebra $g_{\ell}^{0}$ and the center $c_{\ell}$. The absolute irreducibility implies that $c_{\ell}$ is either $0$ or $\mathbb{Q}_{\ell}\mathrm{Id}$. In both cases $V_{\ell}$ is an absolutely irreducible representation of $g_{\ell}^{0}$; in particular, $g_{\ell}^{0}\ne \{0\}$. The semisimplicity of $g_{\ell}^{0}$ implies that
$$g_{\ell}^{0}\subset \mathrm{sl}(V_{\ell})\cong \mathrm{sl}_2(\mathbb{Q}_{\ell}).$$ Now it follows easily that $g_{\ell}^{0}= \mathrm{sl}(V_{\ell})$, because no proper Lie subalgebras of $\mathrm{sl}_2$ are semisimple.

$\newcommand{\g}{\mathfrak{g}}\newcommand{\sl}{\mathfrak{sl}}$The crucial point in the proof is that the absolute Galois group $G_K$ acts irreducibly on $V_{\ell}$, which is based on a nontrivial Shafarevich finiteness theorem (Sect. 1.4). Applying this result to all finite algebraic extensions of $K$, one gets that the $\ell$-adic Lie algebra $\g_{\ell}$ of the image acts irreducibly (and faithfully) on $V_{\ell}$.

So, $\g_{\ell}$ is an irreducible Lie subalgebra of $\mathrm{End}_{\mathbb{Q}_{\ell}}(V_{\ell})$, whose centralizer consists of scalars $\mathbb{Q}_{\ell}\mathrm{Id}$, i.e., the natural faithful 2-dimensional representation of $\g_{\ell}$ in $V_{\ell}$ is absolutely irreducible. Hence, $\g_{\ell}$ is reductive, i.e., splits into a direct sum $$\g_{\ell}=\g_{\ell}^{0}\oplus c_{\ell}$$ of a semisimple Lie algebra $\g_{\ell}^{0}$ and the center $c_{\ell}$. The absolute irreducibility implies that $c_{\ell}$ is either $0$ or $\mathbb{Q}_{\ell}\mathrm{Id}$. In both cases $V_{\ell}$ is an absolutely irreducible representation of $\g_{\ell}^{0}$; in particular, $\g_{\ell}^{0}\ne \{0\}$. The semisimplicity of $\g_{\ell}^{0}$ implies that
$$\g_{\ell}^{0}\subset \sl(V_{\ell})\cong \sl_2(\mathbb{Q}_{\ell}).$$ Now it follows easily that $\g_{\ell}^{0}= \sl(V_{\ell})$, because no proper Lie subalgebras of $\sl_2$ are semisimple.

The crucial point in the proof is that the absolute Galois group $G_K$ acts irreducibly on $V_{\ell}$, which is based on a nontrivial Shafarevich finiteness theorem (Sect. 1.4). Applying this result to all finite algebraic extensions of $K$, one gets that the $\ell$-adic algebra $g_{\ell}$ of the image acts irreducibly (and faithfully) on $V_{\ell}$.

So, $g_{\ell}$ is an irreducible Lie subalgebra of $\mathrm{End}_{\mathbb{Q}_{\ell}}(V_{\ell})$, whose centralizer consists of scalars $\mathbb{Q}_{\ell}\mathrm{Id}$, i.e., the natural faithful 2-dimensional representation of $g_{\ell}$ in $V_{\ell}$ is absolutely irreducible. Hence, $g_{\ell}$ is reductive, i.e., splits into a direct sum $$g_{\ell}=g_{\ell}^{0}\oplus c_{\ell}$$ of a semisimple Lie algebra $g_{\ell}^{0}$ and the center $c_{\ell}$. The absolute irreducibility implies that $c_{\ell}$ is either $0$ or $\mathbb{Q}_{\ell}\mathrm{Id}$. In both cases $V_{\ell}$ is an absolutely irreducible representation of $g_{\ell}^{0}$; in particular, $g_{\ell}^{0}\ne \{0\}$. The semisimplicity of $g_{\ell}^{0}$ implies that
$$g_{\ell}^{0}\subset \mathrm{sl}(V_{\ell})\cong \mathrm{sl}_2(\mathbb{Q}_{\ell}).$$ Now it follows easily that $g_{\ell}^{0}= \mathrm{sl}(V_{\ell})$, because no proper Lie subalgebras of $\mathrm{sl}_2$ are semisimple.

The crucial point in the proof is that the absolute Galois group $G_K$ acts irreducibly on $V_{\ell}$, which is based on a nontrivial Shafarevich finiteness theorem (Sect. 1.4). Applying this result to all finite algebraic extensions of $K$, one gets that the $\ell$-adic Lie algebra $g_{\ell}$ of the image acts irreducibly (and faithfully) on $V_{\ell}$.

So, $g_{\ell}$ is an irreducible Lie subalgebra of $\mathrm{End}_{\mathbb{Q}_{\ell}}(V_{\ell})$, whose centralizer consists of scalars $\mathbb{Q}_{\ell}\mathrm{Id}$, i.e., the natural faithful 2-dimensional representation of $g_{\ell}$ in $V_{\ell}$ is absolutely irreducible. Hence, $g_{\ell}$ is reductive, i.e., splits into a direct sum $$g_{\ell}=g_{\ell}^{0}\oplus c_{\ell}$$ of a semisimple Lie algebra $g_{\ell}^{0}$ and the center $c_{\ell}$. The absolute irreducibility implies that $c_{\ell}$ is either $0$ or $\mathbb{Q}_{\ell}\mathrm{Id}$. In both cases $V_{\ell}$ is an absolutely irreducible representation of $g_{\ell}^{0}$; in particular, $g_{\ell}^{0}\ne \{0\}$. The semisimplicity of $g_{\ell}^{0}$ implies that
$$g_{\ell}^{0}\subset \mathrm{sl}(V_{\ell})\cong \mathrm{sl}_2(\mathbb{Q}_{\ell}).$$ Now it follows easily that $g_{\ell}^{0}= \mathrm{sl}(V_{\ell})$, because no proper Lie subalgebras of $\mathrm{sl}_2$ are semisimple.