Exterior power of Hodge structures

Question

Let $V$ be a $\mathbb{Q}$-vector space and suppose there is a decomposition of $V_{\mathbb{C}}:=V \otimes_{\mathbb{Q}} \mathbb{C}$ into two $\mathbb{C}$-sub-vector spaces i.e., $V_{\mathbb{C}} \cong V^{1,0} \oplus V^{0,1}$, where $V^{i,j}$ are sub-spaces of $V_{\mathbb{C}}$ satisfying $V^{i,j}= \overline{V^{j,i}}$. Note that such a structure is known as a pure Hodge structure of weight $1$. I am looking for an example of $V$ along with such a decomposition such that $$\Bigl(\Bigl({\bigwedge}^2 V^{1,0}\Bigr) \wedge \Bigl({\bigwedge}^2 V^{0,1}\Bigr)\Bigr) \cap {\bigwedge}^4 V \not= {\bigwedge}^2 \Bigl(\Bigl(V^{1,0} \wedge V^{0,1}\Bigr) \cap {\bigwedge}^2 V\Bigr).$$ The question is motivated by the Hodge decomposition for very general abelian varieties, for which the two sides are indeed equal (where $V$ plays the role of the first cohomology of the abelian variety).