  The scale evolution of parton distributions is determined by universal
splitting functions. As a milestone towards the computation of these functions
to four-loop order in QCD, we compute all contributions to the pure-singlet
quark-quark splitting functions that involve two closed fermion loops. The
splitting functions are extracted from the pole terms of off-shell operator
matrix elements, and the workflow for their calculation is outlined. We
reproduce known results for the non-singlet four-loop splitting functions and
validate our new pure-singlet results against fixed Mellin moments.
