This wiki page gives two equivalent definitions for separable algebras.
(1) An associative $K$-algebra $A$ is said to be separable if for every field extension $L/K$ the algebra $ A\otimes_K L$ is semisimple.
and
(2) An algebra $A$ is separable if and only if it is projective when considered as a left module of $A\otimes_K A^{op}$in the usual way.
How would I go about showing that these two definitions are equivalent?
EDIT (An Attempt):
(2)$\implies (1)$ Since $A$ is projective a short exact sequence $$ M\to N\to A\to 0 $$ splits, i.e. $N\cong M\oplus A$. We can think of $L/K$ as a module over the field $K$ (e.g $L=\mathbb{C}$ and $K=\mathbb{R}$). Now, the tensor product is right exact so that $$ M\otimes L \to M\otimes L\oplus A\otimes L\to A\otimes L\to 0 $$ is also exact. Hence, $A\otimes L$ is a summand of a free module therefore semisimple apriori.
See this blog post. I don't know a way to prove this that doesn't involve just classifying the separable algebras over a field.