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

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.