Pensieve header: The zero-co Adjoint representation.

The Adjoint action

DistinctQ[is___] := (Sort[{is}]===Union[{is}]);
expr_ // Ad[t_. a[i_,j_]] := Expand[expr /. {
a[k_,j] /; DistinctQ[i,j,k] :> (E^(-t (Subscript[b, i])))a[k,j]+((Subscript[b, k])/(Subscript[b, i]))(1-(E^(-t (Subscript[b, i]))))a[i,j],
a[j,k_] /; DistinctQ[i,j,k] :> (E^(t (Subscript[b, i])))a[j,k]+((Subscript[b, j])/(Subscript[b, i]))(1-(E^(t (Subscript[b, i]))))a[i,k],
a[k_,i] /; DistinctQ[i,j,k]:> a[k,i] + (1-(E^(-t (Subscript[b, i]))))a[k,j] + ((Subscript[b, k])/(Subscript[b, i]))((E^(-t (Subscript[b, i])))-1)a[i,j],
a[j,i] :> (E^(t (Subscript[b, i])))a[j,i] + ((Subscript[b, j])/(Subscript[b, i]))(1-(E^(t (Subscript[b, i]))))a[i,j],
a[k_,l_] /; DistinctQ[i,j,k,l] :> a[k,l],
a[i,k_] :> a[i,k],
a[k_,k_] :> a[k,k],
a[k_,l_] :> TBD[a[i,j],a[k,l]]
}]

Table[a[i,j]->SeriesCoefficient[a[i,j] // Ad[ϵ a[1,2]], {ϵ,0,1}] // Simplify , {i,3}, {j,3}] // MatrixForm

The semigroup property

Table[a[i,j], {i,3}, {j,3}] // Ad[x a[1,2]] // Ad[y a[1,2]] // FullSimplify // MatrixForm

Table[a[i,j], {i,3}, {j,3}] // Ad[(x +y) a[1,2]] // FullSimplify //  MatrixForm

Table[a[i,j] -> (
(a[i,j] // Ad[x a[1,2]] // Ad[y a[1,2]])==(a[i,j] // Ad[(x +y) a[1,2]]) // Simplify
), {i,3}, {j,3}] // MatrixForm

Verifying R3

(R3L=Table[a[i,j], {i,4}, {j,4}] // Ad[a[1,2]] // Ad[a[1,3]] // Ad[a[2,3]] // Simplify) // MatrixForm

(R3R=Table[a[i,j], {i,4}, {j,4}] // Ad[a[2,3]] // Ad[a[1,3]] // Ad[a[1,2]] // Simplify) // MatrixForm

Table[a[i,j] -> (
(a[i,j] // Ad[a[1,2]] // Ad[a[1,3]] // Ad[a[2,3]])==(a[i,j] // Ad[a[2,3]] // Ad[a[1,3]] // Ad[a[1,2]]) // Simplify
), {i,4}, {j,4}] // MatrixForm