Let $(E,\mathscr{E},\mu)$ be a measure space and $P$ be a Markov kernel : $E \times \mathscr{E} \to [0,1]$. For $f \in L^2(\mu)$ define the Dirichlet form
$$\mathcal{E}(f,P):= \langle f, (Id-P)f \rangle_\mu = \frac{1}{2} \int [f(z')-f(z)]^2 \mu(dz) P(z,dz').$$
Now assume $v \in \{-1,+1\}$ and $T_v$ are sub-stochastic kernels for each $v = 1$ and $v = -1$. Also assume we are given free parameters $\rho_{1,-1}(x)$ and $\rho_{-1,1}(x)$, required to satisfy for all $(x,v) \in E$ $0 \le \rho_{v,-v}(x) \le 1- T_v(x,X)$ and $$\rho_{v,-v}(x)-\rho_{-v,v}(x)= T_{-v}(x,X) - T_v(x,X).$$
For a given such $\rho$, define the transition kernel $$P^\rho((x,v);d(y,w)) = \mathbb{I}\{w=v\} [T_v(x,dy) + \delta_x(dy)(1-T_v(x,X) - \rho_{v,-v}(x))] + \mathbb{I}\{w=-v\} \delta_x(dy) \rho_{v,-v}(x).$$
Then given that $Q$ is a linear operator $L^2(\mu) \to L^2(\mu)$ such that $Qf(x,v) = f(x,-v)$, $Q^2 = Id$, for $f \in \mathbb{R}^E$, we have for any $g \in L^2(\mu)$, $$\mathcal{E}(g,P^{\rho_1}Q) - \mathcal{E}(g,P^{\rho_2}Q) = \frac{1}{2} \int \mu(d(x,v)) (\rho_{2, v, -v}(x) - \rho_{1,v,-v}(x))[g(x,v)-g(x,-v)]^2.$$
I cannot figure out how to get this final identity using the given form of the transition kernel and identity for the given Dirichlet form. I would greatly appreciate any help.