In this paper by Evans and Hrushovski: Projective planes in algebraically closed fields, they characterize projective planes in algebraically closed fields. These are coordinated by the skew-fields of quotients of the endomorphism rings of connected one-dimensional algebraic groups.

I'm trying to understand their "Example 2" (which is actually the general construction according to Theorem 3.3.1), in which (the representation of) all the ground set elements are of the form $\alpha_1x_1*\alpha_2x_2*\alpha_3x_3$, where $x_1$, $x_2$, $x_3$ are independent generic elements of a connected one-dimensional algebraic group $(G, *)$ and $\alpha_1,\alpha_2,\alpha_3\in \operatorname{End}(G)$ not all $0$.

I tried understanding it by looking at examples that I know, but I haven't managed. For example (this is their Example 1), if $x_1$, $x_2$, $x_3$ are algebraically independent over the algebraically closed field $\mathbb{K}$, we can look at all the elements $x_1^{m_1}\cdot x_2^{m_2}\cdot x_3^{m_3}$ with $m_1,m_2,m_3\in \mathbb{Z}$ not all $0$. This example (extended to general rank) shows that matroids linear over $\mathbb{Q}$ are algebraic in all characteristics (?).

Second example, we look at all the elements $\alpha_1 x_1+\alpha_2 x_2+\alpha_3 x_3$ with $\alpha_1,\alpha_2,\alpha_3\in \mathbb{K}$ not all $0$. This example (extended to general rank) is usually used to show that every linear matroid is algebraic over the same field.

If I understand correctly, then in the first case $G=\mathbb{G}_m(\overline{\mathbb{K}(x_1,x_2,x_3)})$ with $\operatorname{End}(G)=\{x\mapsto x^m\ |\ m\in \mathbb{Z}\}$, while in the second case $G=\mathbb{G}_a(\overline{\mathbb{K}(x_1,x_2,x_3)})$ and the endomorphisms in the example are $x\mapsto \alpha x$. If $\operatorname{char}(\mathbb{K})=0$ these are all (?), but otherwise in the second case there are more because $x\mapsto x^p$ is also an endomorphism (?).

I guess my first question is if I understood correctly so far.

My main question is regarding the general construction, because there is another case they describe in the paper (other than $G=\mathbb{G}_m$ and $G=\mathbb{G}_a$), where $G$ is an elliptic curve. I don't understand how the elements look in this case, i.e., how the multiplication and isogenies work here to generate a general element. I'm not even sure what a generic element of $G$ means here—is it a uniformizer for the curve?

I'd really appreciate it if someone could help me understand this example in the general form, so that I will manage to build an example with an elliptic curve.