Loewner variation of the Dirichlet Green function (${\partial }_t{G}_{D_t}(z,w)=-4\mathrm{Im}(1/f_t(z))\mathrm{Im}(1/f_t(w))$)

With half-plane capacity $2t$, $f_t=g_t-W_t$ and $G_{\mathbb{H}}(z,w)=\log |(z-\overline{w})/(z-w)|$, differentiating $G_{D_t}(z,w)=G_{\mathbb{H}}(f_t(z),f_t(w))$ gives the displayed identity. The translation by the driver cancels in the Green function. This rank-one covariance loss balances the quadratic covariation of the harmonic mean in the SLE4 coupling with a Gaussian free field.