  In this paper, we prove that every invertible $2$-local or local automorphism
of a simple generalized Witt algebra over any field of characteristic $0$ is an
automorphism. In particular, every $2$-local or local automorphism of Witt
algebras $W_n$ is an automorphism for all $n\in \mathbb{N}$. But some simple
generalized Witt algebras indeed have $2$-local (and local) automorphisms that
are not automorphisms.

Rings and Algebras