I currently have a field extension $L/K$ that is normal, and I want to prove that its subset of separable elements over $K$ is itself a normal extension of $K$.
In particular, I believe I need to prove that:
1: $J$ is a field and specifically a field extension of $K$.
2: $J/K$ is a normal extension.
$1$ is trivial as long as you are willing to use some of the machinery of field extensions you should have already built in your course. More precisely, we know a field extension is separable if and only if it is generated by separable elements. Therefore if you have $\alpha,\beta\in J$, then it suffices to show that $K(\alpha,\beta)/K$ is separable since $\alpha+\beta,\alpha\beta\in K(\alpha,\beta)/K$, etc. But this is immediate from the above fact if $\alpha,\beta$ are separable by assumption.
For $2$, you need to show that if $f\in K[x]$ is irreducible and has a root in $J$, then it splits in $J$. If we call this root $\alpha$, then $\alpha\in J$ means it is separable over $K$, but the definition of separability of an element is that $\alpha$ is separable over $K$ if and only if its minimal polynomial over $K$ is separable. So we deduce $f$ is separable. But then where should all of its roots lie?