This paper presents the even-N moments (N ≤ 20) of the fourth-order (N³LO) quark-to-gluon splitting function $ P_{\mathrm{gg}}^{(3)}(x) $ in perturbative QCD. These moments, derived from analytically computing off-shell operator matrix elements for a general gauge group, agree with known results and improve upon previous approximations. The results are used to construct more accurate approximations for $ P_{\mathrm{gg}}^{(3)}(x) $, which are now sufficiently precise for most collider-physics applications. The remaining uncertainties of this function are negligible at momentum fractions $ x > 0.1 $. The resulting errors of the convolution of $ P_{gg} $ at $ N^{3}LO $ with a typical quark distribution are small at $ x \gtrsim 10^{-3} $ and exceed 1% only at $ x \lesssim 10^{-4} $ for a strong coupling $ \alpha_{s} = 0.2 $. The paper also discusses the computation of the $ N^{3}LO $ radiative corrections in QCD, which are essential for precision physics at the Large Hadron Collider (LHC) and the future Electron-Ion Collider. The results for the $ N^{3}LO $ splitting functions are presented, including the even moments of $ P_{\mathrm{ik}}^{(3)}(x) $, which have been computed to N=12 for ik=qq and to N=10 for other entries. These results agree with known results from structure functions in deep-inelastic scattering (DIS). The paper also discusses the computation of the $ N^{3}LO $ anomalous dimensions, which are essential for the evolution of parton distribution functions (PDFs). The results for the $ N^{3}LO $ anomalous dimensions are presented, including the even-N moments of the anomalous dimensions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the quark-to-gluon case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the flavour-singlet case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the flavour-singlet case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agreeThis paper presents the even-N moments (N ≤ 20) of the fourth-order (N³LO) quark-to-gluon splitting function $ P_{\mathrm{gg}}^{(3)}(x) $ in perturbative QCD. These moments, derived from analytically computing off-shell operator matrix elements for a general gauge group, agree with known results and improve upon previous approximations. The results are used to construct more accurate approximations for $ P_{\mathrm{gg}}^{(3)}(x) $, which are now sufficiently precise for most collider-physics applications. The remaining uncertainties of this function are negligible at momentum fractions $ x > 0.1 $. The resulting errors of the convolution of $ P_{gg} $ at $ N^{3}LO $ with a typical quark distribution are small at $ x \gtrsim 10^{-3} $ and exceed 1% only at $ x \lesssim 10^{-4} $ for a strong coupling $ \alpha_{s} = 0.2 $. The paper also discusses the computation of the $ N^{3}LO $ radiative corrections in QCD, which are essential for precision physics at the Large Hadron Collider (LHC) and the future Electron-Ion Collider. The results for the $ N^{3}LO $ splitting functions are presented, including the even moments of $ P_{\mathrm{ik}}^{(3)}(x) $, which have been computed to N=12 for ik=qq and to N=10 for other entries. These results agree with known results from structure functions in deep-inelastic scattering (DIS). The paper also discusses the computation of the $ N^{3}LO $ anomalous dimensions, which are essential for the evolution of parton distribution functions (PDFs). The results for the $ N^{3}LO $ anomalous dimensions are presented, including the even-N moments of the anomalous dimensions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the quark-to-gluon case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the flavour-singlet case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agree with known results from structure functions in DIS. The paper also discusses the computation of the $ N^{3}LO $ splitting functions for the flavour-singlet case, including the even-N moments of the splitting functions, which have been computed to N=22. These results agree