Summary: It looks like I'll finally get a good understanding of the gamma notation for moving proper velocities to lab velocities and back. It'll be nice to know it inside and out, but a little irksome given all that can be done with the hyperbolic notation we're not using. I want to maintain my fluency in both. There may be a subtle second notation for inverted Lorentz transforms. As it turns out, the subtle notation difference of moving around indices in the top and the bottom with spaces is meant to keep track of which index comes first when you go back to side by side notation. First, we cover Lorentz transforms, (which are not in fact tensors), and contractions and arrive at the interesting result in equation 1.99: $\Lambda^\mu_\rho \Lambda^\sigma_\mu T^\rho_\sigma = \delta^\sigma_\rho T^\rho_\sigma$ Which indicates the transpose of the Lorentz transform times itself follows a sort of orthogonality rule making use of contravariant indices. Q: ...