Representation of a semisimple Lie algebra onto $\mathfrak{sl}_2(\mathbb C)$

Question

Let $\mathfrak g$ be a complex semisimple Lie algebra and let $\alpha:\mathfrak g\to \mathfrak{sl}_2(\mathbb C)$ be a representation onto the simple algebra $\mathfrak{sl}_2(\mathbb C)$. How to show that there is an ideal $\mathfrak h$ of $\mathfrak g$ such that $\mathfrak h$ is isomorphic to $\mathfrak{sl}_2(\mathbb C)$.

Answer 1

Hint 1: More generally, if $\mathfrak{g}$ is semisimple and $\alpha: \mathfrak{g} \twoheadrightarrow \mathfrak{s}$ is a surjective Lie algebra homomorphism to any Lie algebra $\mathfrak{s}$, then there is an ideal $\mathfrak{h}$ in $\mathfrak{g}$ that is isomorphic to $\mathfrak{s}$.

Hint 2: Namely, $\ker(\alpha)$ is an ideal in $\mathfrak{g}$. That $\mathfrak{g}$ is semisimple means that there is some ideal $\mathfrak{h}$ in $\mathfrak{g}$ such that ...