Suppose that $L$ is a finite Galois extension of the field $K$. If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$ where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ and ${\rm Aut}_{L_2}(L)$ by standard results in Galois theory. ${\rm Aut}_K(L)$ is the group of automorphisms of $L$ fixing $K$.)
I wonder if the following analog result for transcendental field extensions is true. Suppose that $L$ is a finitely generated field extension of a field $K$ and both fields have characteristic 0. Suppose that $L_1$ and $L_2$ are subfields of $L$ containing $K$ that are algebraically closed within $L$. Is it true that ${\rm Der}_{L_1\cap L_2}(L)$ is the Lie algebra generated by ${\rm Der}_{L_1}(L)$ and ${\rm Der}_{L_2}(L)$? (Here ${\rm Der}_K(L)$ is the Lie algebra of derivations of $L$ over $K$.)
