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

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 $$(1)$$?