  We suggest a simple algebraic approach to fix the elements of the $\{ \beta
\}$-expansion for renormalization group invariant quantities, which uses
additional degrees of freedom. The approach is discussed in detail for N$^2$LO
calculations in QCD with the MSSM gluino -- an additional degree of freedom. We
derive the formulae of the $\{ \beta \}$-expansion for the nonsinglet Adler
$D$-function and Bjorken polarized sum rules in the actual N$^3$LO within this
quantum field theory scheme with the MSSM gluino and the scheme with the second
additional degree of freedom. We discuss the properties of the $\{ \beta
\}$-expansion for higher orders considering the N$^4$LO as an example.
