Module-Valued 2-Local Derivations on Reductive Lie Algebras

Let $\text{F}$ be an algebraically closed field of characteristic zero, $\text{g}=\text{s}\oplus \text{z}$ a finite-dimensional reductive Lie algebra over $\text{F}$, and $V$ an arbitrary finite-dimensional $\text{g}$-module. We classify all 2-local derivations of $\text{g}$ on $V$, and show that every 2-local derivation is a derivation if and only if $\dim \text{z}\le 1$ or $V^{\text{g}}=0$. If $\dim \text{z}\ge 2$ and $V^{\text{g}}\mathrm{\ne }0$, the nonlinear homogeneous maps give all exceptional 2-local derivations.