Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be constructed, and then a construction given in Milnor and Hussemoller's book which only applies in the case that the elementary divisors are all 1.

In my case, I have a rank 4 lattice of with a symplectic form of type (1, n), and I have some elements which span an index n sublattice. I would like to somehow relate them to a symplectic basis on the whole lattice, but it seems to me that I would need to have a constructive method for creating such a basis for this to be of any use whatsoever.