Take the following definition:
"A *parabolic subgroup* of a linear algebraic group defined over $\mathbb{C}$ is a subgroup, closed in the Zariski topology, for which the quotient space is a projective algebraic variety."


My questions are:


(i) Why include *closed* in the definition?


(ii) What is an example of a projective algebraic variety that is the quotient of a linear algebraic group by a non-Zariski-closed subgroup?


(iii) What is an example of a quotient by a parabolic subgroup that is not a flag manifold? 


(iv) Elliptic curves cannot be described as quotients of linear algebraic groups. What are other examples of families of varieties that cannot be expressed in this form?



Questions Suggested by the Parabolic Subgroup Definition
If $G={\mathbb C}$ is the additive group and $L\subset G$ is a lattice, then $L$ is not
Zariski closed, $G/L$ exists (in a sense of analytic geometry) and is projective. But of course
$L$ is not a parabolic subgroup of $G$.


