2-dimensional irreducible representation of semisimple Lie algebras

Question

Let $\mathfrak g$ be a complex semisimple Lie algebra. Suppose that $\mathfrak g$ has a $2$-dimensional irreducible representation. How to show that $\mathfrak g$ splits as a direct sum $\mathfrak g_1\oplus\mathfrak g_2$ where $\mathfrak g_1\cong \mathfrak{sl}_2(\mathbb C)$?

Answer 1

Since $\mathfrak{g}$ is semisimple, it is a direct sum of simple ideals $\mathfrak{g}=\mathfrak{g}_1\oplus \cdots \oplus \mathfrak{g}_s$. By looking at the minimal dimension of a nontrivial irreducible representation of the simple Lie algebras $\mathfrak{g}_i$, we see that the dimension can be $2$ only for $\mathfrak{sl}_2(\Bbb C)$, see here, the table on page $3$. Hence one of the factors equals $\mathfrak{sl}_2(\Bbb C)$. For the irreducible representations of direct sums of simple Lie algebras see here:

What are the irreducible representations of a direct sum of Lie Algebras?