Does the carré du champ operator associated with the generator of a contractive $C^0$-semigroup on a Hilbert space have the diffusion property?

Question

Let $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with generator $(\mathcal D(A),A)$, $\mathcal A$ be a subspace of $\mathcal D(A)$ with $fg\in\mathcal A$ for all $f,g\in\mathcal A$ and $$\Gamma(f,g):=\frac12\left(A(fg)-fAg-gAf\right)\;\;\;\text{for }f,g\in\mathcal A.$$ Assume that $T(t)$ is self-adjoint for all $t\ge0$ and $(\mathcal D(A),A)$ is self-adjoint.

Are we able to show that $$\Gamma(fg,h)=f\Gamma(g,h)+g\Gamma(f,h)\tag1$$ for all $f,g,h\in\mathcal A$?

If not, are there mild conditions on the objects which allow us to show $(1)$?