  We present a proof of the factorization of renormalization scheme in
one-loop-corrected Fierz identities. This scheme factorization facilitates the
simultaneous transformation of operator basis and renormalization scheme using
only relations between physical operators; the evanescent operators in the
respective bases may be chosen entirely independently of each other. The
relations between evanescent operators in the two bases is automatically
accounted for by the corrected Fierz identities. We illustrate the utility of
this result with a two-loop anomalous dimension matrix computation using the
Naive-Dimensional Regularization scheme, which is then transformed via one-loop
Fierz identities to the known result in the literature given in a different
basis and calculated in the Larin scheme. Additionally, we reproduce results
from the literature of basis transformations involving the rotation of
evanescent operators into the physical basis using our method, without the need
to explicitly compute one-loop matrix elements of evanescent operators.
