Pensieve header: Experiments with RC, JA, and J; much borrowed from pensieve://Projects/KBH, and hopefully will be returned.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2013-02"];

We load FreeLie.m:

<< FreeLie.m
$SeriesShowDegree=3;
$SeriesCompareDegree = 6;

Some set theoretic definitions:

Domain[f_List] := First /@ f;
f_∖key_ := DeleteCases[f, key -> _];
f_ ∖ keys_List := Fold[#1∖#2&, f, keys];
f1_List ≡ f2_List := Domain[f1]===Domain[f2] && (And @@ (
((#/.f1)≡(#/.f2))& /@ Domain[f1]
));
(* LieDerivation[der_][f_List] := MapAt[LieDerivation[der], f, {All, 2}]; *)
LieMorphism[mor_][f_List] := MapAt[LieMorphism[mor], f, {All, 2}];
M /: M[λ1_, ω1_]⋃ M[λ2_, ω2_] := M[λ1 ⋃ λ2, ω1+ω2];
M[λ1_, ω1_] ≡ M[λ2_, ω2_] := (λ1≡λ2) && (ω1≡ω2);

tm[u_,v_,w_][λ_List] := λ //  LieMorphism[⟨u⟩->⟨w⟩, ⟨v⟩->⟨w⟩];
tm[u_,v_,w_][M[λ_, ω_]] := LieMorphism[⟨u⟩->⟨w⟩, ⟨v⟩->⟨w⟩] /@ M[λ, ω];
hm[x_,y_,z_][λ_List] := Union[λ∖{x,y}, {z->BCH[x/.λ, y/.λ]}];
hm[x_,y_,z_][M[λ_, ω_]] := M[λ // hm[x,y,z], ω];
RC[u_, λx_LieSeries, ub_][ser_] := StableApply[
LieMorphism[⟨u⟩ -> Ad[λx][⟨ub⟩]],
ser
];
RC[u_, λx_LieSeries][ser_] := ser // RC[⟨u⟩, λx,⟨"υ"⟩] // LieMorphism[⟨"υ"⟩->⟨u⟩];
J[u_, λx_] := Module[{s},
IntegrateCWSeries[
div[⟨u⟩,λx // RC[⟨u⟩, s λx]] // LieMorphism[⟨u⟩ -> Ad[-s λx][⟨u⟩]],
{s,0,1}
]
];
tha[u_, x_][λ_List] := MapAt[RC[⟨u⟩, x /. λ], λ, {All, 2}];
tha[u_, x_][M[λ_, ω_]] := M[λ // tha[u,x], (ω+J[⟨u⟩,x /. λ]) // RC[⟨u⟩, x /. λ]];
dm[a_,b_,c_][μ_] := μ // tha[⟨a⟩,b] // tm[⟨a⟩,⟨b⟩,⟨c⟩] // hm[a,b,c];
(SuperPlus[R])[u_,x_] := M[{x->MakeLieSeries[⟨u⟩], u->MakeLieSeries[0]}, MakeCWSeries[0]];
(SuperMinus[R])[u_,x_] := M[{x->MakeLieSeries[-⟨u⟩], u->MakeLieSeries[0]}, MakeCWSeries[0]];