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This doesn't seem to work well in Sage, at least with a naïve approach:

sage: F.<J0,J1,J2> = FreeAlgebra(QQ)
sage: def comm(x,y): return x*y-y*x
sage: I = F.ideal(comm(J0, J1) - J2, comm(J0, J2) + J1, comm(J1, J2) - 2*J0)
sage: A = F.quotient(I)

sage: comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1)
J0^2*J1 - 2*J0*J1*J0 + J1*J0^2 - J1^2*J2 + 2*J1*J2*J1 - J2*J1^2
sage: A(comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1))
J0bar^2*J1bar - 2*J0bar*J1bar*J0bar + J1bar*J0bar^2 - J1bar^2*J2bar + 2*J1bar*J2bar*J1bar - J2bar*J1bar^2

This is clearly flawed. It can't even tell that the generators of the ideal are sent to zero in the quotient:

sage: [A(x)==0 for x in I.gens()]
[False, False, False]

This doesn't seem to work well in Sage, at least with a naïve approach:

sage: F.<J0,J1,J2> = FreeAlgebra(QQ)
sage: def comm(x,y): return x*y-y*x
sage: I = F.ideal(comm(J0, J1) - J2, comm(J0, J2) + J1, comm(J1, J2) - 2*J0)
sage: A = F.quotient(I)

sage: comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1)
J0^2*J1 - 2*J0*J1*J0 + J1*J0^2 - J1^2*J2 + 2*J1*J2*J1 - J2*J1^2
sage: A(comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1))
J0bar^2*J1bar - 2*J0bar*J1bar*J0bar + J1bar*J0bar^2 - J1bar^2*J2bar + 2*J1bar*J2bar*J1bar - J2bar*J1bar^2

This is clearly flawed. It can't even tell that the generators of the ideal are sent to zero in the quotient:

sage: [A(x)==0 for x in I.gens()]
[False, False, False]

This Lie algebra approach:

sage: d = {('J0', 'J1'): {'J2': 1}, ('J0', 'J2'): {'J1': -1}, ('J1', 'J2'): {'J0': 2}}
sage: F.<J0, J1, J2> = LieAlgebra(QQ, d)
sage: F
Lie algebra on 3 generators (J0, J1, J2) over Rational Field
sage: F.bracket(J0, J1)
J2
sage: F.bracket(J0, F.bracket(J0, J1))
-J1
sage: F.bracket(J0, F.bracket(J0, J1)) + F.bracket(F.bracket(J1, J2), J1)
-J1 + 2*J2

Flawed approach below; feel free to ignore (or work on Sage to improve things!).

The following naïve approach doesn't seem to work well in Sage, at least with a naïve approach:Sage:

sage: F.<J0,J1,J2> = FreeAlgebra(QQ)
sage: def comm(x,y): return x*y-y*x
sage: I = F.ideal(comm(J0, J1) - J2, comm(J0, J2) + J1, comm(J1, J2) - 2*J0)
sage: A = F.quotient(I)

sage: comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1)
J0^2*J1 - 2*J0*J1*J0 + J1*J0^2 - J1^2*J2 + 2*J1*J2*J1 - J2*J1^2
sage: A(comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1))
J0bar^2*J1bar - 2*J0bar*J1bar*J0bar + J1bar*J0bar^2 - J1bar^2*J2bar + 2*J1bar*J2bar*J1bar - J2bar*J1bar^2

This is clearly flawed. It can't even tell that the generators of the ideal are sent to zero in the quotient:

sage: [A(x)==0 for x in I.gens()]
[False, False, False]

One Another approach would be to construct the algebra from scratch. You should be able to describe a vector space basis for it, so you can then construct it using CombinatorialFreeModule, the way Clifford algebras (among many other examples) are constructed in Sage.