The nilpotent part M of the Borel of the "(Subscript[U, ϵ])[(Subscript[sl, 3])]" algebra illustrates a bug in the definition of the non-commutative multiplication **. Mathematica fails to simplify the highlighted expression during the evaluation, resulting in an infinite loop.

Initialization + DeclareAlgebra (as in Verification.nb with $k=1)

The “degree carrier / filtration parameter” is ℏ, and all “coupling constants” are proportional to it.

$p=8; $k=1; (* $k can't be ∞ at least because of Faddeev-Quesne. *)
If[$k==0, ϵ=0, ϵ /: (ϵ^(k_ ))/; k>$k :=0]; (* $k=0 fails in Series[..{ϵ,...}] *)
SetAttributes[{SS, SST}, HoldAll];
TRule={(Subscript[T, i_])-> (E^(ℏ (Subscript[t, i]))),T-> (E^(ℏ t))};
SS[ℰ_] := Block[{ℏ,ϵ}, (* Shielded Series *)
Collect[Normal@Series[ℰ, {ℏ,0,$p}],ℏ,Together] ];
SST[ℰ_] := Block[{ℏ,ϵ},Collect[Normal@Series[ℰ/.TRule, {ℏ,0,$p}],ℏ,Together] ];
Simp[ℰ_, op_]:= Collect[ℰ, _CU|_QU, op];
Simp[ℰ_]:=Simp[ℰ,Collect[Normal@Series[#, {ℏ,0,$p}],ℏ,Expand]&];
SimpT[ℰ_]:= Collect[ℰ, _CU|_QU, Collect[Normal@Series[#/.TRule, {ℏ,0,$p}],ℏ,Expand]&];

Differential polynomials (DP):

(Subscript[DP, α_->(Subscript[D, x_]),β_->(Subscript[D, y_])])[P_][λ_]:=Total[CoefficientRules[P,{α,β}]/.({m_,n_}->c_):>c D[λ,{x,m},{y,n}]]

DeclareAlgebra

Unprotect[NonCommutativeMultiply]; Attributes[NonCommutativeMultiply]={};
(NCM=NonCommutativeMultiply)[x_] := x;
NCM[x_,y_,z__] := (x**y)**z;
0**_=_**0=0;
(x_Plus)**y_:=(#**y)&/@x; x_**(y_Plus):=(x**#)&/@y;
B[x_,x_]=0; B[x_,y_] := x**y-y**x;

DeclareAlgebra[U_Symbol, opts__Rule] := Module[{gp,sr,g,cp,CE,pow,k=0,
gs=Generators /. {opts}, cs=Centrals /. {opts} },
((Subscript[#, U])=U@#)&/@gs;
gp=Alternatives@@gs; gp=gp|(Subscript[gp, _]);(* gens *)
sr=Flatten@Table[{g->++k, (Subscript[g, i_])->{i,k}},{g,gs}]; (* sorting -> *) 
cp=Alternatives@@cs; (* cents *)
CE[ℰ_]:= Collect[ℰ, _U, (Expand[#] /. (ℏ^(d_))/;d>$p:>0)&];
(Subscript[U, i_])[ℰ_] := ℰ /. {t:cp:>(Subscript[t, i]),u_U :> Replace[u, x_ :> (Subscript[x, i]),1]};
(Subscript[U, i_])[NCM[]]=pow[ℰ_,0]=U@{}=(Subscript[1, U])=U[];
B[U@(Subscript[(x_), i_]),U@(Subscript[(y_), i_])]:=B[U@(Subscript[x, i]),U@(Subscript[y, i])]=(Subscript[U, i])@B[U@x,U@y];
B[U@(Subscript[(x_), i_]),U@(Subscript[(y_), j_])] /; i=!=j := 0;
B[U@y_,U@x_]:=CE[-B[U@x,U@y]];
x_**(Subscript[1, U]):=x; (Subscript[1, U])**x_:=x;
(a_.*x_U)**(b_.*y_U) := If[a b === 0, 0, CE[a b (x**y)]];
U[xx___,x_] ** U[y_,yy___] := If[OrderedQ[{x,y}/.sr],U[xx,x,y,yy], U@xx**(U@y**U@x+B[U@x,U@y])**U@yy];
U@{c_.*(((l:gp))^(n_)),r___}/;FreeQ[c, gp]:= CE[c U@Table[l,{n}]**U@{r}];
U@{c_.*l:gp,r___} := CE[c U[l]**U@{r}];
U@{c_,r___}/;FreeQ[c, gp]:= CE[c U@{r}];
U@{l_Plus, r___} := CE[U@{#,r}& /@ l];
U@{l_,r___} := U@{Expand[l],r};
U[ℰ_NonCommutativeMultiply]:=U/@ℰ;
(Subscript[𝕆, U])[specs___,poly_] := Module[{sp,null,vs,us},
sp=Replace[{specs}, l_List:>(Subscript[l, null]), {1}];
vs=Join@@(First /@ sp);
us=Join@@(sp /. (Subscript[l_, s_]) :> (l/.(Subscript[x_, i_]):>(Subscript[x, s])));
CE[Total[
CoefficientRules[poly, vs] /. (p_->c_) :> c U@((us^p))
]] /. (Subscript[x_, null]):>x];
pow[ℰ_,n_]:=pow[ℰ,n-1]**ℰ;
(Subscript[𝕊, U])[ℰ_,ss___Rule] := CE@Total[
CoefficientRules[ℰ,First/@{ss}]/.(p_->c_):>c NCM@@MapThread[pow,{Last/@{ss},p}]];
(Subscript[m, j_->k_])[c_.*u_U] := CE[(c/.(((Subscript[#, j])->(Subscript[#, k]))&/@cs))DeleteCases[u,(Subscript[_, (j|k)])]**U@@Cases[u,(Subscript[w_, j]):>(Subscript[w, k])]**U@@Cases[u,(Subscript[_, k])]];
(Subscript[S, i_])[c_.*u_U]:= CE[(c/.(Subscript[S, i])[U,Centrals])DeleteCases[u,(Subscript[_, i])]**(Subscript[U, i])[NCM@@Reverse@Cases[u,(Subscript[x_, i]):>S@U@x]]] ]

Implementing M

DeclareAlgebra[M, Generators ->{X,Y,Z}, Centrals -> {}];
B[(Subscript[Y, M]),(Subscript[X, M])]=-ϵ M@Z;
B[(Subscript[Z, M]),(Subscript[X, M])]= -ϵ M[X,X,Y] ;
B[(Subscript[Z, M]),(Subscript[Y, M])]=-ϵ M[X,Y,Y];

ϵ (Subscript[Y, M])**M[X,Y,Y]