Pensieve header: Solving for V alone (with Kuno). Continued pensieve://People/Kuno.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO4"];
<< FreeLie.m;
<< AwCalculus.m;
Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
α=LS[{x,y}, αs]; β=LS[{x,y}, βs];γ=CWS[{x,y}, γs];
V=Es[⟨x->α, y->β⟩, γ];
R4Eqn=V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V ;
βs[x]= 1/2; βs[y]=0; γs[___]=0;
SeriesSolve[{α, β,γ}, ((ℏ^(-1))R4Eqn) ];
V

V@{8} // Timing

V@{10} // Timing