Recall that Weyl's theorem says that any finite dimensional representation of semi simple Lie algebra is completely reducible. I'm trying some examples to understand this theorem properly.
Since $\bf sl_2(\bf C)$ acts on $V=\bf C^2$ by matrix multiplication, $V$ is an $\bf sl_2$-module.
Decompose $V \otimes V$ as $\bf sl_2$-module.
Since I've just learned about Weyl's theorem so please don't use any high machinery. Any idea or reference is appreciated.