Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\otimes v_1,v_2\otimes v_2,\dotsc,v_n\otimes v_n$.
It seems clear that, for $n\geq 2$, if the square spans of two bases $v=\{v_1,\dotsc,v_n\}$, $v'=\{v_1',\dotsc,v_n'\}$ have non-trivial intersection, then the two bases have proper non-trivial subsets spanning the same space. What happens given three bases $v$, $v'$ and $v''$ such that their square spans, taken together, span a subspace of $V\otimes V$ of dimension less than $3\dim(V)$? What can one say about $v$, $v'$ and $v''$ then?
