magic of Pauli-matrices

in the last two weeks CQW asked about the generation of rotations and Lorentz-transforms from the off-diagonal Pauli-matrices \(\sigma^{(1)}\) and \(\sigma^{(2)}\). what type of transformation would be generated by \(\sigma^{(0)}\) and \(\sigma^{(3)}\)? what types of transformations would be commutative?