I define "standard gram matrix" as the gram matrix g that minimizes the deviation from Toeplitz and that satisfies abs(inv(g)) = abs(g).

I have obtained gram matrices for D4, E8, A15+ with small deviations from Toeplitz that satisfy abs(inv(g)) = abs(g) but I cannot be sure that I have obtained the global optimum. My method for solving the optimization problem specified by the definition of "standard Gram matrix" is ad hoc and I cannot be sure that it has found the global optimum.

I seem to be able to uniquely define standard gram matrices for D4, E8, A15+ for instance.

THIS IS MY AD HOC METHOD FOR DETERMINING THE STANDARD GRAM MATRIX ACCORDING TO MY DEFINITION:

I find a gram matrix g for which abs(inv(g)) - abs(g) is zero. g seems to be unique up to permutations and sign flips, but I am not sure. I choose g to have the most positive signs. inv(mat) is the matrix inverse of mat, and abs(mat) is the absolute values of the matrix elements.

Furthermore I permute the basis vectors to make the gram matrix maximally Toeplitz.

That seems to uniquely define the standard gram matrices for D4 and E8.

For instance, for the 24-cell D4 = D4* lattice I obtain a Toeplitz gram matrix

I wonder whether I have found the maximally Toeplitz gram matrix for E8 and A15+.

For A15+ the permutations are irrelevant. In this case I obtain a gram matrix ga15plus satisfying

I seem to obtain unique standard gram matrices for D4 and E8 and 15 equivalent standard gram matrices for A15+. But I am not sure that I have obtained the most Toeplitz gram matrices for E8 and A15+.

I define "standard Gram matrix" as the Gram matrix g that minimizes the deviation from Toeplitz and that satisfies abs(inv(g)) = abs(g).

I have obtained Gram matrices for D4, E8, A15+ with small deviations from Toeplitz that satisfy abs(inv(g)) = abs(g) but I cannot be sure that I have obtained the global optimum. My method for solving the optimization problem specified by the definition of "standard Gram matrix" is ad hoc and I cannot be sure that it has found the global optimum.

I seem to be able to uniquely define standard Gram matrices for D4, E8, A15+ for instance.

This is my ad-hoc method for determining the standard Gram matrix according to my definition:

I find a Gram matrix g for which abs(inv(g)) - abs(g) is zero. g seems to be unique up to permutations and sign flips, but I am not sure. I choose g to have the most positive signs. inv(mat) is the matrix inverse of mat, and abs(mat) is the absolute values of the matrix elements.

Furthermore I permute the basis vectors to make the Gram matrix maximally Toeplitz.

That seems to uniquely define the standard Gram matrices for D4 and E8.

For instance, for the 24-cell D4 = D4* lattice I obtain a Toeplitz Gram matrix

I wonder whether I have found the maximally Toeplitz Gram matrix for E8 and A15+.

For A15+ the permutations are irrelevant. In this case I obtain a Gram matrix ga15plus satisfying

I seem to obtain unique standard Gram matrices for D4 and E8 and 15 equivalent standard Gram matrices for A15+. But I am not sure that I have obtained the most Toeplitz Gram matrices for E8 and A15+.

-- What do you think of my definition of "standard Gram matrix"?

-- Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?

I find a gram matrix g for which abs(inv(g)) - abs(g) is zero. The unordered pair (g,inv(g)) seems to be unique up to permutations and sign flips, but I am not sure. I choose g to have the most positive signs. inv(mat) is the matrix inverse of mat, and abs(mat) is the absolute values of the matrix elements.

That uniquely defines two gram matrices g for D4 and E8. I choose g to contain more positive signs than inv(g), and thereby obtain a unique standard Gram matrix g for D4 and E8.

-- What do you think of my definition of "standard Gram matrix"?

-- Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?

-- How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?

I find a gram matrix g for which abs(inv(g)) - abs(g) is zero. g seems to be unique up to permutations and sign flips, but I am not sure. I choose g to have the most positive signs. inv(mat) is the matrix inverse of mat, and abs(mat) is the absolute values of the matrix elements.

That seems to uniquely define the standard gram matrices for D4 and E8.

-- How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? ​Do you have any literature references in which "standard Gram matrix" is defined for any reason?