  We study the beta functions for four-dimensional conformal gravity using two
different parametrizations of metric fluctuation, linear split and exponential
parametrization. We find that after imposing the traceless conditions, the beta
functions are the same in four dimensions though the dependence on the
dimensions are quite different. This indicates the universality of these
results. We also examine the beta functions in general quadratic theory with
the Einstein and cosmological terms for exponential parametrization, and find
that it leads to results for beta functions of dimensionful couplings different
from linear split, though the fact that there exists nontrivial fixed point
remains the same and the fixed points also remain the same.
