Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that

$$ F(P \times Q) = H(F(P),F(Q),P,Q)$$

for all $P, Q \in \mathbb{R}[X]$. Two obvious solutions for $F$ and $H$ are

• $F = I$, the identity function, and $H(u,x,y,z) = \alpha u x + \beta u z + \gamma yx + \delta yz$ where $\alpha,\beta,\gamma,\delta \in \mathbb{R}$ and $\alpha +\beta+\gamma+\delta = 1$;

• $F = D$, the derivative, and $H(u,x,y,z) = u z + x y$, from the Leibniz rule.

Do there exist solutions $F$ and $H$ in which $F$ is not a linear combination of $I$ and $D$?