I want to prove the statement : if $K\le E\le F$ and $F$ is separable over $E$, then $P\subset E$.
Here, $P=\{u\in F:\text{$u$ is purely inseparable over $K$}\}\le F$.
I claim that $PE$ is both separable and purely inseparable over $E$, so that $PE=E$ shows $P\subset E$. I proved the first one, but I even don't know whether the second one is true.