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Quotients like this are not well implemented in Sage, and in particular, there is no simple way to compute a basis. Examples of ways in which the implementation is not great:

sage: A2 = SteenrodAlgebra(profile=(3,2,1))
sage: s1 = A2.Sq(1)
sage: I = A2.ideal(s1)
sage: I
Twosided Ideal (Sq(1)) of sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1]
sage: s1 in I
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

....

NotImplementedError: 
sage: R = A2.quotient(I)
sage: R(s1) == 0
False