I heard that the first counter example for the Kac's question "Can we hear the shape of drums?" is given by John Milnor, as two 16-dimensional tori that are isospectral but not isometric. I found the original 1-page paper of Milnor that explains why two tori are isospectral. Also, we can show that by using the theory of theta functions and modular forms, which is explained in detail in Don Zagier's book "123 of modular forms".
However, I can't find the reference for non-isometricity of two tori. I found that two tori are $E_8 \oplus E_8$ and $\Lambda_{16}$ (where the latter one is called Barnes-Wall lattice), but I'm not sure how to show that two corresponding tori aren't isometric. (In fact, it is hard to describe the latter lattices explicitly.) Could anyone provide a brief explanation or a good reference for this? Thanks in advance.