There is some simplification, similar to Lagrange's identity, for the multiplication of double summation ?
Double Summation:
$\left( \sum\limits_{\substack{m=1}}^N \sum\limits_{\substack{n=1}}^N a_{mn} \right) \left( \sum\limits_{\substack{m=1}}^N \sum\limits_{\substack{n=1}}^N b_{mn} \right)$
Lagrange's identity:
$\left( \sum\limits_{\substack{m=0}}^N a_{m} \right) \left( \sum\limits_{\substack{m=0}}^N b_{m} \right) = \sum\limits_{\substack{m=0}}^{2N} \sum\limits_{\substack{k=0}}^{m} a_kb_{m-k} - \sum\limits_{\substack{m=0}}^{N-1}\left(a_m\sum\limits_{\substack{k=N+1}}^{2N-m}b_k+b_m\sum\limits_{\substack{k=N+1}}^{2N-m}a_k \right)$