Local automorphisms on finite-dimensional Lie and Leibniz algebras

We prove that a linear mapping on the algebra $\mathfrak{sl}_n$ of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form $\mathfrak{sl}_n\dot +\mathcal{I}$ is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$ with $\dim \mathcal{L} \geq 3$ which admit local automorphisms which are not automorphisms.