  In the paper, we make a detailed study on the generalized Crewther Relation
(GCR) between the Adler function ($D$) and the Gross-Llewellyn Smith sum rules
coefficient ($C^{\rm GLS}$) by using the newly suggested single-scale approach
of the principle of maximum conformality (PMC). The resultant GCR is
scheme-independent, whose residual scale dependence due to unknown higher-order
terms are highly suppressed. Thus a precise test of QCD theory without
renormalization scheme and scale ambiguities can be achieved by comparing with
the data. Moreover, a demonstration of the scheme independence of commensurate
scale relation up to all orders has been presented. And as the first time, the
Pade approximation approach has been adopted for estimating the unknown $5_{\rm
th}$-loop contributions from the known four-loop perturbative series.
