When doing an exercise about linear representations of finite groups I stumbled upon this Isomorphism in the comments of another post which I was not aware of.
In this context $V$ and $W$ are finite dimensional Vector complex vector spaces. I was already able to show that the dimension of the two Vector spaces or rather Tensors are the same. As such it should be sufficient to construct an injective or surjective map between them.
However I tried to construct one such as: $$ \phi: (v_1 \oplus w_1) \cdot (v_2\oplus w_2) \in Sym^2(V \oplus W) \mapsto (v_1 \cdot v_2 ) \oplus ((v_1+v_2) \otimes (w_1 + w_2)) \oplus (w_1 \cdot w_2) \in Sym^2(V)\oplus (V\otimes W)\oplus Sym^2(W) $$ However it seems to me as if this map is not injective nor surjective since $\phi((v_1 \oplus w_1) \cdot (v_2\oplus w_2)) = \phi((v_1 \oplus w_2) \cdot (v_2\oplus w_1))$.
Any tips for constructing such a map or should I be trying a different approach?