Let $k$ be a field of characteristic zero and $V_1, V_2$ two finite dimensional $k$-vector spaces. I would like to show that
$$ \wedge ^2 (V_1 \otimes V_2) = (\operatorname{Sym} ^2(V_1) \otimes \wedge^2(V_2)) \oplus (\wedge^2(V_1) \otimes \operatorname{Sym} ^2(V_2))$$
My idea: I want to embed $\operatorname{Sym} ^2(V_1) \otimes \wedge^2(V_2)$ and $\wedge^2(V_1) \otimes \operatorname{Sym} ^2(V_2)$ naturally into $\wedge ^2 (V_1 \otimes V_2)$, show that their intersection is zero and use dimension count.
The second step on zero intersection works if we realize $\operatorname{Sym} ^2(V_1) \otimes \wedge^2(V_2)$ and $\wedge^2(V_1) \otimes \operatorname{Sym} ^2(V_2)$ as subspces of $V_1^{\oplus 2} \otimes V_2^{\oplus 2}$ and use that for general space $V$ the $\operatorname{Sym} ^2(V)$ and $\wedge^2(V)$ intersect trivially.
The third step, the dimension count works fine as well.
Problem: How to embed $\operatorname{Sym} ^2(V_1) \otimes \wedge^2(V_2)$ and $\wedge^2(V_1) \otimes \operatorname{Sym} ^2(V_2)$ naturally into $\wedge ^2 (V_1 \otimes V_2)$. Is there an canonical way to do it or should one do something "exotical" to obtain the embeddings?
ps: I know that there is advanced machinery to solve it using Schur functors & Young symmetrizers, but I'm curious if in this case it's possible to solve it with elemenary methods. The only aspect that I still cannot manage yet is the embedding issue...
For a vector space $V$, one may realize $\wedge^2V\subset V\otimes V$ as the subspace generated by elements of the form $v\otimes w-w\otimes v$ with $v,w\in V$, and similarly $Sym^2V\subset V\otimes V$ as the subspace generated by elements of the form $v\otimes w+w\otimes v$.
Now, there is a map \begin{align} Sym^2V_1\otimes\wedge^2V_2&\to \wedge^2(V_1\otimes V_2)\\(v_1\otimes w_1+w_1\otimes v_1)\otimes (v_2\otimes w_2-w_2\otimes v_2)&\mapsto v_1\otimes v_2\otimes w_1\otimes w_2-v_1\otimes w_2\otimes w_1\otimes v_2\\&+w_1\otimes v_2\otimes v_1\otimes w_2-w_1\otimes w_2\otimes v_1\otimes v_2. \end{align} One may also rephrase these maps in terms of $\wedge^2V$ and $Sym^2V$ as quotients of $V\otimes V$ (by the subspaces generated by $v\otimes w+w\otimes v$ and $v\otimes w-w\otimes v$, respectively): \begin{align} Sym^2V_1\otimes\wedge^2V_2&\to \wedge^2(V_1\otimes V_2)\\(v_1\cdot w_1)\otimes (v_2\wedge w_2)&\mapsto (v_1\otimes v_2)\wedge(w_1\otimes w_2)-(v_1\otimes w_2)\wedge(w_1\otimes v_2). \end{align}
The construction for the other map is similar.