Pensieve header: A $\nu^{-1}$-like computation, except with the middle strand punctured, as suggested by Zsuzsi ... version of July 2018.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO3"];
<< "../WKO4/FreeLie.m";
<< "../WKO4/AwCalculus.m";
$SeriesShowDegree=4;

Φs[2,1]=Φs[3,1]=Φs[3,2]=0;Φs[3,1,2]=1/24; (Subscript[Φ, 0])=DKS[3, Φs];
SeriesSolve[(Subscript[Φ, 0]), (((Subscript[Φ, 0])^(σ[3,2,1]))≡-(Subscript[Φ, 0]))∧((Subscript[Φ, 0])**((Subscript[Φ, 0])^(σ[1,23,4]))**((Subscript[Φ, 0])^(σ[2,3,4]))≡((Subscript[Φ, 0])^(σ[12,3,4]))**((Subscript[Φ, 0])^(σ[1,2,34])))];
(Subscript[Φ, 0])@{6}

DK2Es[s___][ζ_] := El[ζ // αMap[s], CWS[0]] // Γ;
DK2Es[1,2,3][(Subscript[Φ, 0])]

νinv=(Subscript[Φ, 0]) //DK2Es[1,2,3] // dS[2] // dm[3,2,2] // dm[2,1,x]

νinv@{6}

zν=(Subscript[Φ, 0]) //DK2Es[1,2,3] // dS[2] //tη[2] // dm[3,2,2] // dm[2,1,x];
zν@{6}

(zν**νinv**zν**νinv)@{6}

(νinv // dΔ[x,x,y] // dm[x,y,x])@{6}

zν1=(Subscript[Φ, 0]) //DK2Es[1,2,3] // dS[1] //tη[2] // dm[1,2,1] // dm[1,3,x];
zν1@{6}

(zν1**νinv)@{6}

zν1u=(Subscript[Φ, 0]) //DK2Es[1,2,3] // dS[1] // dm[1,2,1] // dm[1,3,x];
zν1u@{6}

zν2=(Subscript[Φ, 0]) //DK2Es[1,2,3] // dS[1] //tη[2] // dS[3] //dm[1,2,1] // dm[1,3,x];
zν2@{6}

(zν1u**zν2)@{6}

ϕ=LS[Sum[(Subscript[Φ, 0])[k], {k,6}][[2]]]

((Subscript[J, 1])[ϕ] // LieMorphism[LW@1->LW@1, LW@2->LW@1])@{6}

(j[⟨LW[1]->ϕ, LW[2]->0⟩]// LieMorphism[LW@1->LW@1, LW@2->-LW@1])@{6}

ϕ1=LieMorphism[LW[1]->-LW[1]][ϕ]

((Subscript[J, 1])[ϕ1] // LieMorphism[LW@1->LW@1, LW@2->-LW@1])@{6}

(j[⟨LW[1]->ϕ1, LW[2]->0⟩]// LieMorphism[LW@1->LW@1, LW@2->LW@1])@{6}