Im sure this is a beginners question.

Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states:

"If $\varphi$ is any anisotropic class in $I^n(k)$, then $rank(\varphi) \geq 2^n$."

It is well known that for the kernel of the discriminant $ker(e_1) = I^2$ holds.

Since there is a filtration $W(k) \supseteq I(k) \supseteq I^2(k) \supseteq..$ , the set of classes of elements in $I(k)$ with non trivial discriminant is the complement $I(k)\backslash I^2(k)$. Since the Arason-Pfister-Hauptsatz only gives a lower bound on the rank of elements in $I(k)$ it might be possible that there is a class $\varphi \in I(k)\backslash I^2(k)$ with $rank(\varphi) = 4$ or even higher.

Do you have an example such that this happens?

In "Galois Cohomology" Serre referres to some paper of Alexander Merkurjev, in which he shows that it is possible for "every" even $N\geq1$ to find a $k$ having $u$-invariant $u(k)=N$ ,such that $I^3(k)$ vanishes i.e. $k$ has cohomological depth $cd(k)=2$. This is basically the answer to an analog of my question in case of finding some anisotropic rank $8$ class in $I^2(k)\backslash I^3(k)$.