Local automorphisms of finite dimensional simple Lie algebras

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