Beauzamy and Dégot's Identity

For $P$, $Q$, $R$, and $S$ Polynomials in $n$ variables

$$[P\cdot Q, R\cdot S]= \sum_{i_1,\ldots,i_n\geq 0} {A \over i_1!\cdots i_n!},$$

where

$$A\equiv[R^{(i_1,\ldots,i_n)}(D_1,\ldots,D_n)Q(x_1,\ldots,x_n)P^{(i_1,\ldots,i_n)}(D_1,\ldots,D_n)S(x_1,\ldots,x_n)]$$

$D_i=\partial/\partial x_i$ is the Differential Operator, $[X, Y]$ is the Bombieri Inner Product, and

$$P^{(i_1,\ldots,i_n)}=D_1^{i_1}\cdots D_n^{i_n} P.$$

See also Reznik's Identity