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

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

Exists it, others, no trivial solutions for $H$ ?