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Ref of the original paper https://arxiv.org/abs/1911.00892. The biggest issue for me is they don't have proper upper/lower indices. A lower index is a leg pointing down, while an upper index is a leg pointing up. \delta should be \delta_i^j. The upper/lower I like to think of as extra type information. You can pair covectors and vectors. The other issue is with the dot product of course, I would write a_i b^i.

They're sneaking the Euclidean metric tensor in everywhere, which means there's no distinction between upper and lower indices. I agree it's confusing, and I'd rather they kept tensor orientations intact.

The dot product is double contraction with the Euclidean metric; your notation is contraction of a vector and a dual vector.

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