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I guess i have to guess. Do we need the following simple data extraction?!

sage: L = lie_algebras.three_dimensional_by_rank(QQ, 3, names=['E','F','H'])
sage: A = L.pbw_basis()
sage: E, F, H = A.algebra_generators()
sage: expression = E^3*F^2 - F^3*E^2

sage: expression
PBW['E']^3*PBW['F']^2 - PBW['E']^2*PBW['F']^3 + 6*PBW['E']*PBW['F']^2*PBW['H'] - 6*PBW['E']*PBW['F']^2 - 6*PBW['F']*PBW['H']^2 + 6*PBW['F']*PBW['H']

sage: for monomial, coeff in expression:
....:     print "Coeff=%3s Monomial=%s" % (coeff, monomial)
....:     
Coeff=  6 Monomial=PBW['F']*PBW['H']
Coeff=  1 Monomial=PBW['E']^3*PBW['F']^2
Coeff=  6 Monomial=PBW['E']*PBW['F']^2*PBW['H']
Coeff= -6 Monomial=PBW['F']*PBW['H']^2
Coeff= -6 Monomial=PBW['E']*PBW['F']^2
Coeff= -1 Monomial=PBW['E']^2*PBW['F']^3

sage: fixed_monomial = F*H^2
sage: for monomial, coeff in expression:
....:     if monomial == fixed_monomial:
....:         print "Coeff=%3s for fixed monomial=%s" % (coeff, fixed_monomial)
....:      

sage: for monomial, coeff in expression:
....:     if A(monomial) == A(fixed_monomial):
....:         print "Coeff=%3s for fixed monomial=%s" % (coeff, fixed_monomial)
....:         
....:     
Coeff= -6 for fixed monomial=PBW['F']*PBW['H']^2
sage:

The above sample code illustrates some ways to proceed and some traps to avoid.