 the identities $$\begin{aligned} \mathrm d\mathrm{BCH}_\mathfrak p^0(x,y)&=\langle \left[y,\omega_0(x,y)\right],\partial_y\mathrm{BCH}_\mathfrak p^0(x,y)\rangle,\\ \mathrm d D_\mathfrak p^0(x,y)&=\langle \left[y,\omega_0(x,y)\right],\partial_y D_\mathfrak p^0(x,y)\rangle+\