Lemma 7.4 is wrong as stated: the rank of D^T D can be smaller
than the rank of D. For example take the field C of complex
numbers and the matrix  D^T = (1 i) of rank 1. We have D^T D = 0. 
The result can be corrected by working with the real numbers, in
which case the restriction of the form to a complement of the
kernel of D is non-singular, since it is positive definite there.
This is sufficient for our purposes since the matrix D to which
this result is applied has integer entries.