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About that morphism $$F(z,w,s,t)=2iz-iw+3\sqrt3s-\sqrt3t\ ,$$ seen as a morphism of abelian groups, it can be defined as follows as a morphism from $\Bbb Z^4$ to the (abelian group obained by applying the forgetful functor from the) field $\Bbb Q(\sqrt{-1},\sqrt 3)$ as follows, using the short cuts $j=\sqrt{-1}$ and $b=\sqrt 3$:

A = ZZ^4
F.<j,b> = NumberField([sqrt(-1).minpoly(), sqrt(3).minpoly()])

f = A.Hom(F^1)((2*j, -j, 3*b, -b))

Then we have for instance:

sage: f((1,1,1,1))
(j + 2*b)

The "trick" is best explained by examining the following objects:

sage: F
Number Field in j with defining polynomial x^2 + 1 over its base field
sage: F^1
Vector space of dimension 1 over Number Field in j with defining polynomial x^2 + 1 over its base field