Conventions on August 18

Only one algebra (not necessarily Lie!) at a time, and it comes with a list of (Latin) generators and their Greek duals, and a default PBW order. If some Verma module is needed, it is added once and by means of extra generators.
Generators are of the form (Subscript[x, α,β])[i], (Subscript[a, α])[i], etc., where i is the tensor factor.
In the commutative context we have (Subscript[𝔼, i,j->k])[(Subscript[L, 0]),(Subscript[L, 1]),...], where (Subscript[L, p]) is the coefficient of (ϵ^p), and (maybe) (Subscript[𝕆, i,j->k])[F].
In the non-commutative context, generators do not carry a “tensor factor” designations. Instead we consider expressions like (Subscript[U, i])[x,y,y,x].

Issues: Upper and lower?

Experiments

(E^a)(E^b)

Conventions on August 17

Generators can be any Symbol (e.g. x,y,z,a,b,t,x12...). We use Format and Interpretation to make the appearances prettier, as in x12->(Subscript[x, 12]). 
Latin to Greek duality is global: (SuperStar[x])=ξ, etc.
In the non-commutative context, generators do not carry a “tensor factor” designations. Instead we consider expressions like
(Subscript[U1, i])[x,y,y,x]

Experiments

U1@i[x,y,y,x]

Format[x12,StandardForm]=(Subscript[x, 1,2])

{x13@3,(Subscript[x12, 3])}

Do[
Format[Evaluate@Symbol["x"<>ToString[i]<>ToString[j]], StandardForm]=(Subscript[x, 10i+j]),
{i,3},{j,3}
]

(Subscript[x23, 4])

Module[{x},
(Subscript[x, i_,j_]):=Symbol["x"<>ToString[i]<>ToString[j]];
Do[
Format[(Subscript[x, i,j]),StandardForm]=ToExpression[SubscriptBox["x",ToString@i<>ToString@j]],
{i,3},{j,3}
];
Do[
B[(Subscript[x, i,j]),(Subscript[x, k,l])]=(Subscript[x, i,l]),
{i,3},{j,3},{k,3},{l,3}
]
]

(Subscript[B[x12,x23], 4])

(Subscript[x, 1,2])

ToBoxes[(Subscript[y, 1,2])]

ToExpression[SubscriptBox["y",RowBox[{1,",","2"}]]]