  We prove that every $2$-local automorphism on a finite-dimensional
semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of
characteristic zero is an automorphism. We also show that each
finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim
\mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an
automorphism.
