I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true when the Pythagoras number of $K$ is $1$, and clearly false when it is $3$, but I am not sure what can happen when it is $2$. The two properties I am looking at are:

- The torsion subgroup of $W(K)$ has exponent (at most) $2$.
- For any $a,b\in K^\times$ which are sums of squares, the Pfister form $\langle\langle a,b\rangle\rangle$ is hyperbolic.

When $K$ is Pythagorean, 1. is true because $W(K)$ is either $\mathbb{Z}/2\mathbb{Z}$ or is torsion-free, and 2. is obvious since $a$ and $b$ are actually squares by hypothesis.

When the Pythagoras number is at least $3$, we can choose some $a\in K^\times$ which is a sum of squares, but not of $2$ squares. Then $\langle \langle a,a\rangle\rangle = \langle \langle -1,a\rangle\rangle$ is not hyperbolic, which already shows that 2. is false. But $a$ is positive in any real closure of $K$, so $\langle \langle a\rangle\rangle$ becomes hyperbolic in any real closure, and thus by Pfister's local-global theorem $\langle \langle a\rangle\rangle$ is a torsion element in $W(K)$. Since $2\langle \langle a\rangle\rangle = \langle \langle -1,a\rangle\rangle\neq 0$ in $W(K)$, this shows that 1. is also false.

Are those properties [true/false/it depends] when the Pythagoras number is $2$?