Reference for Noam Elkies comments on integral lattices and fundamental parallelotope ORIGINAL: I would like to find some more detailed references to explain  2012 comments of Prof. Noam Elkies at https://mathoverflow.net/questions/103152/determinant-of-integer-lattice-basis-of-l-x-1-ldots-x-n-a-1x-1-cdotsa 
Here are screen captures 


The recent questions I answered that got me interested were:
Basis for the kernel of linear map for linear Diophantine equation in three variables
On largest box not containing integer vector solutions
On largest box not containing integer vector solutions-$II$
In case of interest, here is an article on a  recent breakthrough: http://www.ams.org/journals/notices/201702/rnoti-p102.pdf 
 A: August 2018: this is also in the section called "The orthogonal lattice" in the article Merkle-Hellman Revisited: A Cryptanalysis of the Qu-Vanstone Cryptosystem Based on Group Factorizations, by P. Nguyen and J. Stern (1997). Link to a ps version
Found it. I am quite fond of a book by Wolfgang Ebeling called Lattices and Codes. I have the second edition, in which all relevant material is in pages 1-5, and this Proposition 1.2 and its proof are on page 5. In the third edition, the statement of Proposition 1.2 is on page 4, with the proof split on pages 4 and 5. 
Ebeling also points out the proof in Looijenga and Peters(1981). The directly relevant discussion is just pages 153-154, (pdf pages [9],[10]), while this is Lemma (2.3) on page 154. The  section devoted to lattices is pages 153-157, pdf [9]-[13]. Oh, well. For some reason the jpegs from the article are not showing properly. Sigh. Or, having tried one pasting the imgur address in a new tab, it is just not appearing here within the first minute...


