Consider linear $F: \mathbb R[x] \rightarrow \mathbb R[x]$ with : $F(P\times Q)=H(F(P),F(Q),P,Q)$ for some $H$.

A possible solution for $H$ is $H(u,x,y,z)=u\times z+x \times y$, where $F$ is the classical derivate $D$.

Does there exist any other non-trivial* solutions for $H$ ?

*: $F$ isn't a linear combinaison of $Id$ and $D$.