Relative to an orthonormal frame $\{e_i\}_{i=1}^n$, we have $${\operatorname{R}}_{ijkl} = {\operatorname{R}}_{klij} = \langle {\operatorname{R}}(e_i^*\wedge e_j^*), e^*_k\wedge e^*_l\rangle = -2R_{abij}\langle e^*_a\wedge e^*_b, e^*_k\wedge e_l^*\rangle
 = -2R_{ijkl}.$$ Similarly, we will use ${\operatorname{S}}$ for the operator $\nabla {\operatorname{R}}\in  T^*M\otimes{\operatorname{End}}({\wedge^2 T^*M})$ such that $${\operatorname{S}}(X, \eta)_{ij} = -2\nabla_aR_{ijbc}X_a\eta_{bc}$$ and adopt the notation $${\operatorname{S}}_{mijkl} = \langle{\operatorname{S}}(e_m, e_i^*\wedge e_j^*), e^*_k\wedge e^*_l\rangle = -2\nabla_mR_{ijkl}.$$

Now define ${\tilde{{\operatorname{R}}}}$, ${\hat{{\operatorname{R}}}}: {\wedge^2 T^*M}\to{\wedge^2 T^*M}$ by $${\tilde{{\operatorname{R}}}}= {\operatorname{R}}\circ {\tilde{{\operatorname{P}}}}, \quad {\hat{{\operatorname{R}}}}= {\operatorname{R}}\circ{\hat{{\operatorname{P}}}}$$ and ${\tilde{{\operatorname{S}}}}$, ${\hat{{\operatorname{S}}}}\in T^*M\otimes{\operatorname{End}}({\wedge^2 T^*M})$ by $${\tilde{{\operatorname{S}}}}= \nabla {\operatorname{R}}\circ {\tilde{{\operatorname{P}}}}, \quad {\hat{{\operatorname{S}}}}= \nabla{\operatorname{R}}\circ{\hat{{\operatorname{P}}}}.$$ Here, $${\tilde{{\operatorname{S}}}}(X,\eta) = (\nabla_X{\operatorname{R}})\circ {\tilde{{\operatorname{P}}}}(\eta), \quad {\hat{{\operatorname{S}}}}(X, \eta) = (\nabla_X{\operatorname{R}}) \circ {\hat{{\operatorname{P}}}}(\eta),$$ for $X\in TM$ and $\eta\in {\wedge^2 T^*M}$.

\[lem:init\] At $\tau = 0$, we have $$\begin{aligned}
\label{eq:p0}
   & \nabla {\tilde{{\operatorname{P}}}}= \nabla {\hat{{\operatorname{P}}}}= \nabla {\operatorname{F}}= 0,\end{aligned}$$ $$\begin{aligned}
\label{eq:r0}
    &{\operatorname{R}}\circ {\operatorname{F}}= 0, \quad {\hat{{\operatorname{R}}}}= 0, \quad {\tilde{{\operatorname{R}}}}= {\operatorname{R}},\end{aligned}$$ and $$\begin{aligned}
\label{eq:dr0}
    &\nabla {\operatorname{R}}\circ {\operatorname{F}}= 0, \quad {\hat{{\operatorname{S}}}}= 0, \quad {\tilde{{\operatorname{S}}}}= \nabla {\operatorname{R}}.\end{aligned}$$

When $\tau = 0$, we have $\nabla J = 0$ and consequently $$R(X, Y, JZ, W) + R(X, Y, Z, JW) = 0$$ for all $X$, $Y$, $Z$, $W\in TM$. The former implies that $\nabla {\operatorname{F}}= \nabla {\hat{{\operatorname{P}}}}= \nabla {\tilde{{\operatorname{P}}}}= 0$ and the latter implies that ${\operatorname{R}}\circ {\operatorname{F}}= 0$; in fact, $$({\operatorname{R}}\circ {\operatorname{F}})(Z^*\wedge W^*)(X, Y) = 