This post (Composite of two purely inseparable extensions is purely inseparable.) says that if $E < F$ is purely inseparable and $F < K$ is purely inseparable, then $E < K$ is separable.
However, does the converse hold? That is, if $E < K$ is purely inseparable and we have $E < F < K$, are $E < F$ and $F < K$ purely inseparable?
Yes, assume that $K/E$ is purely inseparable and let $F$ be an intermediate field. As $F\subseteq K$ the field extension $F/E$ is trivially purely inseparable. Let $x\in K$. Then the minimal polynomial $f$ of $x$ over $F$ is a divisor of the minimal polynomial of $x$ over $K$. The latter one has by assumption only one root (in some extension field), hence also $f$ has only one root. It follows that $x$ and therefore also $K$ is purely inseparable over $E$.