This is because both techniques transform the same set of 4 numbers to the same new set of 4 numbers using the same variable beta. To see this work in practice, please examine the problem sets. Confirm the interval is unchanged. The first term is conserved as expected. The vector portion of the square is frame dependent.
L is too complex for simple calculations.
Mathematica does the grunge work. In these cases, the boost quaternion L becomes a very simple. Redshifts have a plus instead of the minus. Over 50 problems in a sophomore-level relativistic mechanics class have been solved using quaternions. Problems in special relativity can be solved either using 4-vectors, the Minkowski metric and the Lorentz group, or using quaternions.