$G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $$G$$ a Lie group and let $$V$$ a representations of $$G$$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align}

I have no idea how to prove this fact. Any suggestions? Thanks in advance!

• How about $v\otimes w\mapsto (v\otimes w + w\otimes v, v\otimes w - w\otimes v)$ ? – Max Feb 11 at 18:59
• @Max It works, thank you!! – userr777 Feb 17 at 13:37