Consider the natural action of $W_1=k\left<x,\frac{d}{dx}\right>$ on $X=\mathbb C[x]$, then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentailly a $\mathfrak{sl_2}$-tuple ($[x\frac{d}{dx},\frac{d}{dx}] =-\frac{d}{dx} ,[x\frac{d}{dx},x^2\frac{d}{dx}]= x^2\frac{d}{dx}), [\frac{d}{dx},x^2\frac{d}{dx}]=2x\frac{d}{dx} )$.

So $X$ is a $\mathfrak{sl_2}$-module, and one easily checks $X$ is a non-split extension of $L(-1)$ by $L(1)$, for instance $\mathbb C 1 \subseteq X=\mathbb C[x]$ is invariant and isomorphic to $L(1)$. Here $L(\lambda)$ is the the normalized simple module with highest weight $\lambda-1$.

How to generalize such construction? Can we construct more examples of non-split extension of simple modules of simple lie algebras using differential operators on some good spaces such as symmetric spaces?