Quotient ring involving Laurent polynomials

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Hi all, I have actually two closely related questions:

Q1 Below is a working code that involves no Laurent polynomials:

sage: P = PolynomialRing(ZZ,'t0,t1')
sage: A.<T0,T1> = QuotientRing(P,[t0^2-1, t1^2-1])
sage: T0^2
sage: T1^2

The output is

1
1

I want to make a q-analog of it so it will produce its (Q,q)-deformation:

(Q-1)*T0 + Q
(q-1)*T1 + q

My attempt below didn't compile, and I'm wondering if there's a good solution to it.

sage: R = LaurentPolynomialRing(ZZ,'Q,q')
sage: P = PolynomialRing(R,'t0,t1')
sage: A.<T0,T1> = QuotientRing(P,[t0^2-1, t1^2-1])
sage: T0^2
sage: T1^2

Q2 I'm trying to implement the most general Hecke algebra of type B2 with unequal parameters for the two nodes. This is labeled as a TODO in the Sage Reference Manual (cannot post direct link due too low karma):

doc.sagemath.org/html/en/reference/algebras/sage/algebras/iwahori_hecke_algebra.html

Does anyone know any update to this?