Every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple.

Question

Can I deduce that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple from the fact that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is semisimple? How? If so, how can I use the previous fact to show that every finite-dimensional representation of a simply connected covering $G$ of $SL_2(\mathbb{R})$ factors through a representation of $SL_2(\mathbb{R})$.

Answer 1

The statement that every finite-dimensional representation of a semisimple Lie algebra is semisimple is called Weyl's theorem. Probably the easiest way to answer the first part of your question is to take the algebraic proof of Weyl's theorem, using Casimir elements, and note that every argument there works not only for $\mathbb{C}$, but for any field of characteristic zero. In particular, the result is still true for $\mathbb{R}$.