ON CENTRAL AUTOMORPHISMS OF FREE METABELIAN LIE ALGEBRAS

Let $F_m$ be the free metabelian Lie algebra of rank $m$ over a field $K$ characteristic 0. An automorphism $\varphi$ is called central if commutes with every inner $F_m$. Such automorphisms form centralizer $\text{\rm C}(\text{\rm Inn}(F_m))$ group Inn}(F_m)$ in Aut}(F_m)$. We provide an elementary proof to show that Inn}(F_m))=\text{\rm Inn}(F_m)$.