Pensieve header: The Free Lie / Lazy Evaluation Abstraction Challenge.

MakeLieSeries[ser_Symbol, expr_] := (
ser[] = Hold[MakeLieSeries[ser, expr]];
ser[d_Integer] := ser[d] = Expand[expr /. w_LW /; Deg[w] ≠ d -> 0];
LieSeries[ser]
);

AddLieSeries[ss___LieSeries] := AddLieSeries[ss] = Module[{ser},
ser=Unique[AddLieSeries];
ser[] = Hold[AddLieSeries[ss]];
ser[d_Integer] := ser[d] = Plus @@ ((#[d])& /@ {ss});
LieSeries[ser]
];

b[s1_LieSeries, s2_LieSeries] := b[s1,s2] = Module[{ser},
ser=Unique[b];
ser[] = Hold[b[s1,s2]];
ser[d_Integer] := ser[d] = Sum[
b[s1[k], s2[d-k]],
{k,1,d-1}
];
LieSeries[ser]
];

LieDerivation[der_Symbol, rules_List] := (
der[]=Hold[LieDerivation[der, rules]];
(der[w_LW] /; Deg[w] == 1) := (der[w] = MakeLieSeries[w /. Append[rules, _LW -> 0]]);
der[w_LW] := der[w] = Module[{x,y},
{x,y}=LyndonFactorization[w];
AddLieSeries[b[der[x],y], b[x,der[y]]]
];
der[s_LieSeries] := der[s] = Module[{ser},
ser=Unique[LieDerivationOnLieSeries];
ser[] = Hold[der[s]];
ser[d_] := ser[d] = Sum[
der[s[k]][d],
{k,1, d}
];
LieSeries[ser]
];
der[as_ASeries] := Omitted;
der[cws_CWSeries] := Omitted;
der[expr_][d_] := Expand[expr /. {w_LW :> der[w][d], s_LieSeries :> der[s][d]}];
LieDerivation[der]
);

BCHBase = Module[{bch},
bch=Unique["BCHBase"];
bch[] = Hold[BCHBase];
bch[1] = ⟨"x"⟩+⟨"y"⟩;
bch[d_Integer] := bch[d] = Expand[Plus[
adSeries[E^(-ad), MakeLieSeries[⟨"y"⟩]][MakeLieSeries[⟨"x"⟩]][d],
-adSeries[(1-E^(-ad))/ad-1, LieSeries[bch]][EulerE[LieSeries[bch]]][d]
]/d];
LieSeries[bch]
];

JA[-1, ___] = MakeCWSeries[0];
JA[n_, y_LW, μ_LieSeries, ss_] := JA[n,y,μ,ss] = Module[
{s, sμ, μs},
sμ = ScaleLieSeries[s, μ];
μs=StableApply[LieMorphism[{y -> Ad[ScaleLieSeries[1,sμ]][LW[z]]}], μ];
μs = μs // LieMorphism[{LW[z] -> y}];
IntegrateCWSeries[
AddCWSeries[
JA[n-1, y, μ, s] // LieDerivation[{y -> b[μs, y]}],
div[y, μs ]
],
{s,0,ss}
]
];
JA[y_LW, μ_LieSeries] := JA[y,μ] = Module[{cws,s},
cws=Unique[JA];
cws[]=Hold[JA[y,μ]];
cws[d_Integer] := cws[d] = JA[d-1, y, μ, s][d] /. s->1;
CWSeries[cws]
];