Can someone please help me to prove the following statement:
In $\mathbb{R}^{d+1}$ we consider the Lie algebra generated by the vectors field:
$X_j=∂_{q_j},\hspace{1cm} Y_j=∂_{q_j}V(q)iτ_0,\;j=1,...,d$
where $V$ is a polynomial in $R[x_1,..,x_d]$ of degree less or egal to $r$
then the lie algebra is nilpotent and its order is equal to the degree of $V$.