In the “Rlgroups pdf” file embedded below we present proofs for the expression of n-dimensional rotations and respective Lorentz transformations as exponential functions of linear combinations of matrix generators. The generators of the restricted Lorentz group appear , if we take a Lorentz coordinates transform, changing like a second order contravariant tensor. Finally we show also how the Dirac conserved current transforms under a Lorentz coordinates transformation like a contravariant Lorentz vector.
This chapter you can also find in the post
Notes on … quantum field theory