Random Stuff

PJ[u_, λ_] := J[u, λ] // LieMorphism[u -> Ad[-λ][u]];

$SeriesShowDegree=3;
{u=⟨"u"⟩, w=⟨"w"⟩, λ0= MakeLieSeries[u+w+b[u,w]]}

s λ0

λ0 // CC[s λ0, u]

div[u,λ0 // CC[s λ0, u]]

div[u,λ0 // CC[s λ0, u]] // LieMorphism[u -> Ad[-s λ0][u]]

t1=IntegrateCWSeries[
div[u,λ0 // CC[s λ0, u]] // LieMorphism[u -> Ad[-s λ0][u]],
{s,0,1}
]

J[u, λ0]

t2=(J[u, λ0] // LieMorphism[u -> Ad[-λ0][u]])

t1≡t2

{u=⟨"u"⟩, w=⟨"w"⟩, λ0= MakeLieSeries[u+w+b[u,w]]};
{PJ[u,s λ0], P[u,s λ0]}

div[u,λ0]

Table[
Coefficient[P[u,s λ0]@k, s] == div[u,λ0]@k,
{k,8}
]

P[u, λ0] ≡ PJ[u, λ0]

PJ[u, ϵ λ0 // CC[s λ0, u]]

div[u, λ0 // CC[s λ0, u]]

Table[
Coefficient[PJ[u, ϵ λ0 // CC[s λ0, u]]@k, ϵ] == div[u, λ0 // CC[s λ0, u]]@k,
{k,6}
]

Table[
Coefficient[P[u, ϵ λ0 // CC[s λ0, u]]@k, ϵ] == div[u, λ0 // CC[s λ0, u]]@k,
{k,6}
]

PJ[u,s λ0]

div[u, λ0 // CC[s λ0, u]] //

Even more random

P[u,(s+ϵ) λ0]

t1 = P[u,(s+ϵ) λ0] // CC[s λ0, u]

t2 = P[u,s λ0] // CC[s λ0, u]

ϵ λ0 // CC[s λ0, u]

P[u, ϵ λ0 // CC[s λ0, u]]

t3=P[u, ϵ λ0 // CC[s λ0, u]]

t2+t3

t1≡t2+t3