Given the matrix $\phantom{-} \pmatrix{ 0 & a+b& -b+a&0\\ \phantom{-}a+c& a+d&\phantom{-} b+c&b+d\\ -c+a&c+b&-d+a&d+b\\ 0&c+d&\phantom{-}d-c&0\\ }. $ How could one decompose it into the (smallest) sum of tensor product, like $M\otimes E_{k}$ and $E_{n}\otimes M$, where $M=\pmatrix{a&b\\c&d}$ and $E_j$ is a matrix with entries $0$ or $\pm 1$, like $\pmatrix{0&1\\0&0}$ or $\pmatrix{1&\phantom{-}1\\1&-1}$. I tried it with sums of $M\otimes 1_{kn}$ and $1_{kn}\otimes M$ ($1_{kn}$ has a $1$ at element $(k,n)$ and $0$ elsewhere), but haven't found a solution, yet...
In order to minimise the number of summands, feel free to substitue every element by its negative (if a solution exists).