If  $( A  B | C  D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that 
$\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$?

This arose in connection with some calculations involving theta functions.  There seems to be some indication that it might be true, and I could not come up with a counterexample; but experts in the area may well know the answer right off.  

Thanks for any help.