This question is about euclidean lattices, regular arrays of points in R^N.

Why are there 3 Gram matrices for the E8 lattice? They are not related by a similarity transformation; they have different eigenvalues. How do I simply prove that the three different gram matrices, with different eigenvalues, determine the same lattice? 

Is there a concept for the degeneracy of the Gram matrix, which is 3 for E8, and what is the term for it?  Are there any other examples?  

The eigenvalues of the Gram matrices of other lattices seem unique, at least in the tables quoted below from Gabriele Nebe.  I cannot find any other examples of this.  E8 is the only lattice for which I can find different Gram matrices with different eigenvalues in Nebe's tables.


What property of E8 lattice is it that makes its Gram matrix eigenvalues undetermined?

What is the relationship among the 3 Gram matrices of E8?  Do they actually determine the same lattice?  I will check the depth of their holes by brute force, to see if they are different, but I have not done that yet.

https://math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html

https://math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.b.html

https://math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8_code.html

I think the 3 must be related by translations (and rotations).  That would mean they correspond to a tricoloring of E8 & there are 3 different classes of points in E8, but I can't find that info.  The chromatic number of E8 is 16, not 3.

https://arxiv.org/pdf/1907.09751.pdf