Let $L/K$ be a finite Galois extension of fields with Galois group $G$. There is a well-known isomorphism of $K$-algebras $$L \otimes_K L \to \prod_{\sigma \in G} L$$ given on pure tensors by $a \otimes b \mapsto (a\sigma(b))_\sigma$. This can be found for example in Bourbaki's Algebra V.10.4.
I am struggling to find an explicit description of the inverse isomorphism. I'd appreciate any help.