Let $F$ be a field of characteristic not equal to to $2$, $W(F)$ Witt ring of the quadratic forms. I've been trying to prove that $I^2(F)=0$ implies that every binary quadratic form over $F$ is universal.
EDIT: My original idea does not seem like the right one. After some toying around, it would seem like the following implication holds:
$\langle 1,a,b,ab\rangle$ isotropic for all $a,b\in F^\times$ $\Rightarrow \langle 1,a\rangle$ universal for all $a\in F^\times$.
Does anyone know if this is true or how to prove it?