Pensieve header: The true ${\mathfrak g}^\epsilon$ within $sl_2$ and without; old technology UEA.

((t𝔤)^ϵ) within (Subscript[sl, 2])

ρe=(({{0, 1}, {0, 0}})); ρf=(({{0, 0}, {ϵ, 0}}));  ρh=(({{ϵ, 0}, {0, -ϵ}})); ρg=(({{1, 0}, {0, -1}})); ρ0=(({{0, 0}, {0, 0}}));
MB[x_?MatrixQ, y_?MatrixQ] := x.y-y.x;
Simplify@{MB[ρg,ρe]==2ρe, MB[ρg,ρf]==-2ρf, MB[ρe,ρf]==ρh, ρh-ϵ ρg==ρ0}

ρb=(({{-1, 0}, {0, -1}})); gl2rule={b->ρb,c->((ϵ^(-1))ρb-ρg)/2,u->-ρf, w->ρe};
Simplify[{MB[w,c]==w, MB[c,u]==u, MB[u,w]==b-2ϵ c} /. gl2rule]

MatrixForm /@ Simplify /@ ({w,u,b,c} /. gl2rule)

(b-ϵ c)⊗c+u⊗w /.gl2rule /. a_⊗b_:>MatrixForm/@Simplify/@(a⊗b)

(b-ϵ c)⊗(-c)+(-u)⊗w /. {c->((ϵ^(-1))b-g)/2,u->-f, w->e} /. a_⊗b_:>Simplify/@(a⊗b)

Implementing ((t𝔤)^ϵ)=⟨b,e,g,f⟩/[g,e]=2e, [g,f]=-2f, [e,f]=b+ϵ g, [b,*]=0)

PBWRule={e->1,g->2,f->3};
B[U@g,U@e]=2U@e; B[U@g,U@f]=-2U@f; B[U@e,U@f]=b U[]+ϵ U[g];

$TD=3; ℏ /: (ℏ^(d_. ))/;d>$TD :=0;

x_⪯y_:=OrderedQ[{x,y}/.PBWRule]; x_≺y_:=!OrderedQ[{y,x}/.PBWRule];
Simp[ℰ_]:= Collect[ℰ, _U, Expand];

(Subscript[U, i_])[ℰ_] := ℰ /. {b->(Subscript[b, i]),t->(Subscript[t, i]), u_U :> Replace[u, x_ :> (Subscript[x, i]),1]};
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[x_,x_]=0;
B[U[y_],U[x_]]:=B[U[y],U[x]]=Simp[-B[U[x],U[y]]];
B[x_,y_] := x**y-y**x;

Unprotect[NonCommutativeMultiply];
NonCommutativeMultiply[x_] := x;
0**_=_**0=0;
x_**U[]:=x; U[]**x_:=x;
(a_*x_U)**(b_*y_U) := If[a b === 0, 0, Simp[a b (x**y)]];
(a_*x_U)**y_ := Simp[a(x**y)]; x_**(a_*y_U) := Simp[a(x**y)];
(x_Plus)**y_ := (#**y)& /@ x; x_**(y_Plus) := (x**#)& /@ y;

U[xx___,x_] ** U[y_,yy___] := If[x⪯y,U[xx,x,y,yy], U@xx**(U@y**U@x+B[U@x,U@y])**U@yy];

UU[l___,((x_)^(n_)),r___]:= UU[l,Sequence@@Table[x,{n}],r];
UU[l___,1,r___]:= UU[l,r];
UU[]=U[];
UU[l_,r___]:=U[l]**UU[r];

UProducts[{},0]={UU[]};
UProducts[{},n_Integer] /; n>0 = {};
UProducts[{x_,xs___}, n_Integer] := Sort@Flatten@Table[UU[(x^k)]**u, {k,0,n},{u,UProducts[{xs},n-k]}];
UProducts[xs_List, k_Integer, n_Integer] := UProducts[Flatten@Table[(Subscript[x, j]), {x,xs}, {j,k}], n];
UProducts[any__, {n_}] := Flatten@Table[UProducts[any,k], {k,0,n}];

m[i_,j_,k_][ℰ_] := Simp[ℰ /. {
u_U :> UU@@Join[DeleteCases[u,(Subscript[x_, i|j])],U@@Cases[u,(Subscript[x_, i]):>(Subscript[x, k])],U@@Cases[u,(Subscript[x_, j]):>(Subscript[x, k])]],
(Subscript[b, i|j])->(Subscript[b, k])}]

$Basis={U@e,U@g,U@f};

Table[{x,y}->B[x,y],{x,$Basis}, {y,$Basis}]// MatrixForm

Module[{x,y},Union@Table[{x,y}=t; B[x,y]+B[y,x],{t, Tuples[$Basis, 2]}]]

Module[{x,y,z}, DeleteCases[Table[
({x,y,z}=t)-> B[x,B[y,z]]+B[y,B[z,x]]+B[z,B[x,y]],
{t, Tuples[$Basis, 3]}
], _->0]]

f⊗e+((1/2) (b+g ϵ))⊗((1/2) (g-(b/ϵ)))

(Subscript[r, i_,j_]):=USimp[(Subscript[U, i])[f](Subscript[U, j])[e]+(1/4)(-(ϵ^(-1))δ (Subscript[U, i])[b](Subscript[U, j])[b]+2 (Subscript[U, i])[b](Subscript[U, j])[g]+ϵ (Subscript[U, i])[g](Subscript[U, j])[g])+α((Subscript[U, i])[b](Subscript[U, j])[g]-(Subscript[U, i])[g](Subscript[U, j])[b])];
(Subscript[r, 1,2])

(Subscript[r, 1,2]) /. {δ->0, α->((-1)/4)}

UB[(Subscript[r, 1,2]),(Subscript[r, 1,3])]+UB[(Subscript[r, 1,2]),(Subscript[r, 2,3])]+UB[(Subscript[r, 1,3]),(Subscript[r, 2,3])] // USimp

UB[(Subscript[r, 1,2]),(Subscript[r, 1,3])]

UB[(Subscript[r, 1,2]),(Subscript[r, 2,3])]

UB[(Subscript[r, 1,3]),(Subscript[r, 2,3])]