Simple algebras of Weyl type, II

Over a field $\mathbb{F}$ of any characteristic, for a commutative associative algebra $A$, and for a commutative subalgebra $D$ of $\operatorname{Der}_{\mathbb{F}}(A)$, the vector space $A[D]$ which consists of polynomials of elements in $D$ with coefficients in $A$ and which is regarded as operators on $A$ forms naturally an associative algebra. It is proved that, as an associative algebra, $A[D]$ is simple if and only if $A$ is $D$-simple. Suppose $A$ is $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie algebra, the subquotient $[A[D],{A[D]}\,]/(Z(A[D])\cap [A[D], {A[D]}\,])$ is simple (except for one case), where $Z(A[D])$ is the center of $A[D]$. The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.