Local automorphisms of finite dimensional simple Lie algebras

Let ${\mathfrak g}$ be a finite dimensional simple Lie algebra over an algebraically closed field $K$ of characteristic $0$. A linear map $\varphi:{\mathfrak g}\to {\mathfrak g}$ is called a local automorphism if for every $x$ in ${\mathfrak g}$ there is an automorphism $\varphi_x$ of ${\mathfrak g}$ such that $\varphi(x)=\varphi_x(x)$. We prove that a linear map $\varphi:{\mathfrak g}\to {\mathfrak g}$ is local automorphism if and only if it is an automorphism or an anti-automorphism.