  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.
