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I think the question can be written in a concise way:

Why do Rotation are represented using two upper/lower index ($R_{ij}$), while, for example, Lorentz Transformation involve index up and index down ($\Lambda^{\mu}_{v}$)? Aren't both, essentially, a matrix? So shouldn't the rotation also be written as $R^{\mu}_{v}$?

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    In Euclidean space, the usual Cartesian coordinate systems have metric tensor equal to the identity matrix, so raising and lowering indices has no effect. Therefore, one might as well write all indices as lower. With the Minkowski metric of relativity, the position of indices matters (a minus sign). Even in Euclidean space, if you use non-Cartesian coordinates (like polar coordinates), the metric tensor isn't the identity and it becomes worthwhile to keep track of upper and lower indices. – Andreas Blass Apr 20 '21 at 22:10
  • See my answer here. The location of the indices is important. – K.defaoite Apr 20 '21 at 22:23

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