I remark that the subformula CCCpqCrpCCpqCrq of this single axiom follows from just B', C and W:

       1. CCpqCCqrCpr                                                                [B']
       2. CCpCqrCqCpr                                                                [C]
       3. CCpCpqCpq                                                                     [W]
D2.1 = 4. CCpqCCrpCrq
D4.1 = 5. CCpCqrCpCCrsCqs
D4.3 = 6. CCpCqCqrCpCqr 
D6.5 = 7. CCCpqCrpCCpqCrq

[Line 2 here can be replaced with restricted versions of C (e.g., C¹¹¹ = CCCpsCCqtCruCCqtCCpsCru) and the proof will still go through.  So line 7 is, for example, provable also in EI, which has C⁰¹⁰ = CCpCCqtrCCqtCpr in place of RI's C. Unfortunately, though, its addition to B', C, and I does not axiomatize EI.]

The formula in question, CCCpqCrpCCpqCrq, is of interest because it, in turn, gives W in the presence of B', C, and I:  

        1. CCpqCCqrCpr                                                             [B']
        2. CCpCqrCqCpr                                                             [C]
        3. Cpp                                                                                  [I]
        4. CCCpqCrpCCpqCrq                                                  [The theorem]
D2.1 =  5. CCpqCCrpCrq
D2.3 =  6. CpCCpqq
D4.5 =  7. CCCpqqCCpCpqq
D1.6 =  8. CCCCpqqrCpr
D8.7 =  9. CpCCpCpqq
D2.9 = 10. CCpCpqCpq                                                              [W]

Consequently {B', C, I, CCCpqCrpCCpqCrq} is a new 4-base for RI, one which has the interesting feature of being 3-solvable in the sense of Rezus [On a theorem of Tarski, Libertas Mathematica, vol. 2 (1982), pp. 63-97].  Briefly, a formula α is m-solvable (m any positive integer) just in case the result of using α as major premise and detaching Cpp as minor, then using that result as major and detaching Cpp as minor again, etc., produces Cpp itself after m iterations; and a base for a logic is m-solvable just in case each member of the base is itself k-solvable for some positive integer k less than or equal to m. 
          Rezus originally conjectured that no theorem of RI not provable in BCI is m-solvable for any m.  He abandoned that conjecture in the face of a  counter-example provided by David Meredith (our CCCpqCrpCCpqCrq is another), and subsequently speculated that a solvable basis for RI might exist.  That latter speculation thus proves to be correct.  Incidentally, if we replace C with Pon = CpCCpqq / p→((p→q)→q) in our new 4-base, we get a 2-solvable base for RI.