Let $R$ be a regular local $k$-algebra, where $k$ is a field (of characteristic zero). Let $x_1,\dots,x_n$ be a regular system of parameters, so that $dx_1,\dots,dx_n$ gives a trivialization of the module of differential forms over $R$. Write $R_1=k\langle dx_1,\dots,dx_n\rangle$.
Then $R_1^*$ will naturally be a vector space of derivations on $R$, and in fact $R\otimes_k R_1^*$ will be isomorphic to the module of derivations on $R$.
For $X,Y\in R_1^*$, write $D_X$, $D_Y$ for the derivations on $R$ they give rise to. Then my question is: do we have $[D_X,D_Y]=0$? I.e. do $D_X$ and $D_Y$ commute, as linear maps?
I think it's straightforward to see that the derivation $[D_X,D_Y]$ vanishes on the subalgebra generated by $x_1,\dots,x_n$. However, the $x_i$ do not generate all of $R$, so I'm not sure how to show this derivation vanishes completely (which I suspect is true).