In Patrick Morandi's book Field and Galois Theory Example 4.24 they write Let $k$ be a field of characteristic 2, let $F=k(x, y)$ and $S=F(u)$, where $u$ is a root of $t^2+t+x$, and let $K=S(\sqrt{u y})$. Then $K / S$ is purely inseparable, $S / F$ is separable, so $S$ is the separable closure of $F$ in $K$.
Then they try to show that $I=F$, which will prove that $K / I$ is not separable since $K / S$ is not separable.
Now $K/S$ is purely inseparable cause $\sqrt{uy}$ is purely inseparable over $S$ since $\sqrt{uy}^{2^1}\in S$. And $u$, $u+1$ are the roots of $t^2+t+x$ so $S/F$ is separable.
Here How they got $S$ is the separable closure of $F$ in $K$?