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Very nice. I have a question though. You create a new matrix by summing the generators and orthogonalize this new one. How do you know the transform will then simultaneously orthogonalize each summand? Does this come from the theory?

Hello Dan. Strictly spoken I sum not over generators, but over the whole group. That is the trick that

Conjugate[T[a]] . H . T[a] equals H.

The details are outlined in the text cited.

Correction: it must be of course

TransposeConjugate[T[a]] . H . T[a] equals H.