I am trying to develop a function(s) to do some commutator algebra to compute the enveloping algebra and ideals of a Lie algebra. For example if I have $SU(2)$ algebra generated by $L_i$ ($i=1,2,3$), I would like to evaluate commutators between polynomials of type $L_1^mL_2^nL_3^p$ where $m,n,p$ are integers.
Now I think I should start by defining a commutator function as a mapping which is:
(a) antisymmetric, (b) is a derivation (Leibniz property), and (c) satisfies Jacobi identity.
Since I do not want to plug in explicit realizations for the operators, defining commutator $[A,B] = AB - BA = C$ would not really help. I want to define commutator as a rule which would just assign $[A,B] \rightarrow C$ and when computing $[A,BC] \rightarrow [A,B]C + B[A,C]$ and so on. I tried by defining something with Leibniz property as follows:
Comm[a_**b_, c_**d_] := a**Comm[b, c]**d + a**c**Comm[b, d] + Comm[a, c]**d**b + c**Comm[a, d]**b;
This works fine for evaluating $[a,bcd]$ but it doesn't do anything to $[abc,d]$ for example. So I am wondering what should I do to correctly define this and how would I go about to define this mapping with the desired properties.
commutatorReducethat does the transformations in the direction you want. $\endgroup$