Separable extension of $k \subset K \subset L$

Question

Consider a tower a finite field extension $k \subset K \subset L$. I see that:

i) If $k \subset L$ is separable then $K \subset L$ is separable

ii) If $K \subset L$ and $k \subset K$ separable then $k \subset L$

I can not find any proof to the converse statements neither spot a counterexample. So my question is: do the converses to (i) and (ii) hold?

Answer 1

The converse statement of i) is wrong, of ii) it's true:

conterexample for i): Take $K=L$ and $k \subset L$ arbitrary non separable extension

proof of converse of ii): $K \subset L$ is separable by i) und $k \subset K$ is also separable since by definition & assumption every $a \in K \subset L$ is separable over $k$.