  We consider the supersymmetry (SUSY) transformations in the chiral Lagrangian
for $N = 1$ supergravity (SUGRA) with the complex tetrad following the method
used in the usual $N = 1$ SUGRA, and present the explicit form of the SUSY
trasformations in the first-order form. The SUSY transformations are generated
by two independent Majorana spinor parameters, which are apparently different
from the constrained parameters employed in the method of the 2-form gravity.
We also calculate the commutator algebra of the SUSY transformations on-shell.
