Computing parabolically induced embeddings of semisimple complex Lie algebras in Weyl algebras

An arbitrary proper parabolic subalgebra $\mathfrak{p}$ of a simple complex Lie algebra $\mathfrak{g}$ induces an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n$,andmoregenerallyanembedding${\mathfrak g}\hookrightarrow \mathbb W_n\otimes \operatorname{End}{} V$,where$\mathbb W_n$ is the Weyl algebra in $n$ variables, $n$ is the dimension of the nilradical of ${\mathfrak p}$,and$V$isanarbitrary${\mathfrak p}$-module. We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings $G_2 \hookrightarrow \mathbb W_5$,$F_4 \hookrightarrow \mathbb W_{15}$,$E_6 \hookrightarrow \mathbb W_{16}$,$E_7 \hookrightarrow\mathbb W_{27}$,$E_8 \hookrightarrow \mathbb W_{57}$ arising in this fashion.