Simple Lie algebras in varieties generated by Lie algebras of cartan type

The author proves that if $K$ is the algebra of regular functions of any smooth affine indecomposable algebraic variety ($\operatorname{char}K=0$) then it can be recovered from its Lie algebra of regular vector fields using a certain multilinear polynomial mapping. It is established that if, for some natural number $n$, a finitely generated Lie algebra $\mathscr G$ over an algebraically closed field $K$ ($\operatorname{char}K=0$) satisfies all identities of the Lie algebra $\widetilde W_n(K)$ of all derivations of the power series algebra in $n$ commuting variables, then $\mathscr G$ contains a proper subalgebra of finite codimension; moreover, for any maximal ideal $J$ of $\mathscr G$, either $\dim_K\mathscr G/J\leqslant n^2+2n$ or $\mathscr G/J$ can be embedded in $\widetilde W_n(K)$.
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