Given an extension $L|\mathbb{k}$ I need to prove that the ring $L\otimes_{\mathbb{k}}L$ embeds into a direct product of algebraic extensions of $L$ if and only if $L|\mathbb{k}$ is separable.
I can prove that separability of $L|\mathbb{k}$ entails that $L\otimes_{\mathbb{k}}L$ doesn’t have nilpotent elements. This is due to the fact that for any element $\alpha\in L$ and its minimal polynomial $f_{\alpha}\in \mathbb{k}[X]$ we now that $L[X]/(f_{\alpha})\hookrightarrow L\otimes_{\mathbb{k}} L$. Thus, if $\alpha$ is inseparable then $L(X)/f_{\alpha}$ is not reduced.
However I stuck here. Also i’m not sure if the notion of algebraic extension here is necessary. I would appreciate any advice.