Derivations of Lie Algebras of Vector Fields in Infinitely Many Variables

Let $W_X(\mathbb{F})$ be the Lie algebra of all derivations of the polynomial algebra $\mathbb{F}[X]$ in infinitely many variables. We describe all derivations of $W_X(\mathbb{F})$ over a field of characteristic zero and prove that all such derivations are inner. We also consider the subalgebras $W_\infty(\mathbb{F})$ and $W_{\text{fin}}(\mathbb{F})$ of the algebra $W_X(\mathbb{F})$ and describe all of their derivations.