A remark about the Lie algebra of infinitesimal conformal transformations of the Euclidian space

Infinitesimal conformal transformations of $R^n$ are always polynomial and finitely generated when $n>2$. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over $R^n$, $n>1$, is maximal in the Lie algebra of polynomial vector fields. When $n$ is greater than 2 and $p,q$ are such that $p+q=n$, this implies the maximality of an embedding of $so(p+1,q+1,R)$ into polynomial vector fields that was revisited in recent works about equivariant quantizations. It also refines a similar but weaker theorem by V. I. Ogievetsky.