A field extension of degree $\leq 2$

Question

Inspired by this question, I am asking the following question (with same notations):

Let $K(ab,a+b) \subset K(a,b) \subset L$, where $a,b \in L$ and $K \subseteq L$ are fields of characteristic zero.

We have, $K(ab,a+b) \subset K(a,b)=K(ab,a+b)(a)=K(ab,a+b)(b)$, and the degree of the field extension $K(ab,a+b) \subset K(a,b)$ equals the degree of the minimal polynomial of $a$ over $K(ab,a+b)$ which equals the degree of the minimal polynomial of $b$ over $K(ab,a+b)$, which is $\leq 2$, since $x^2 - (a+b)x + ab \in K(ab,a+b)[x]$ has roots $a$ and $b$. (An extension of degree $\leq 2$ is easily seen to be Galois).

Is there something 'interesting' to say if it happens that $K(ab,a+b) = K(a,b)=K(ab,a+b)(a)=K(ab,a+b)(b)$?

Of course, in this case $a \in K(ab,a+b)$ and $b \in K(ab,a+b)$. By 'interesting' I mean some connection between $a,b,ab,a+b, K$. (Perhaps the answer depends on the transcendence degree of $K(ab,a+b)$ over $K$).

I really apologize if my question seems useless.

Answer 1

Here's a thing: $a \in K(ab,a+b) \iff a^2 \in K(ab,a+b)$.

This is because $a = \frac{a^2 + ab}{a+b} \in K(ab,a+b,a^2)$.