{\operatorname{R}}||{\operatorname{A}}| + C|{\operatorname{R}}|\left(|{\operatorname{B}}| + |{\hat{{\operatorname{R}}}}|\right)\\
 \label{eq:shev}
 \left|\left(D_{\tau} + \Delta\right){\hat{{\operatorname{S}}}}\right| &\leq C|\nabla\nabla{\operatorname{R}}| |{\operatorname{A}}| + C|\nabla{\operatorname{R}}||{\hat{{\operatorname{R}}}}| + C|{\operatorname{R}}|\left(|{\operatorname{B}}| + |{\hat{{\operatorname{R}}}}| + |{\hat{{\operatorname{S}}}}|\right)\end{aligned}$$ for some $C = C(n)$ on $M\times [0, T]$.

The only properties of the projections ${\tilde{{\operatorname{P}}}}$ and ${\hat{{\operatorname{P}}}}$ of which the argument in [@KotschwarRFHolonomy] makes use is that they remain complementary and orthogonal, evolve according to $D_{\tau} {\tilde{{\operatorname{P}}}}= D_{\tau}{\hat{{\operatorname{P}}}}= 0$ (which follows here from the fact that $D_{\tau} J = 0$), and that ${\mathcal{H}}= {\tilde{{\operatorname{P}}}}({\wedge^2 T^*M})$ remains closed under the Lie bracket. The computations actually hinge on the relations $$\label{eq:larelations}
    [{\mathcal{H}}, {\mathcal{H}}] \subset {\mathcal{H}}, \quad [{\mathcal{H}}, {\mathcal{K}}] \subset {\mathcal{K}},$$ implied by this latter condition. We will review here the derivation of , but refer the reader to [@KotschwarRFHolonomy] for the rest of the details.

Recall that, under , the curvature operator evolves according to the equation $$\label{eq:rmev}
  \left(D_{\tau} + \Delta\right){\operatorname{R}}= - {\operatorname{R}}^2 - {\operatorname{R}}^{\#},$$ where the second term denotes the Lie algebra square ${\operatorname{R}}^{\#} = {\operatorname{R}}\#{\operatorname{R}}$. Here, for $\mathrm{M}$, $\mathrm{N} \in {\operatorname{End}}({\wedge^2 T^*M})$, $\mathrm{M}\#\mathrm{N}\in {\operatorname{End}}({\wedge^2 T^*M})$ is the operator defined at $p\in M$ by $$\label{eq:sharpdef}
 (\mathrm{M}\# \mathrm{N})(\eta) = \frac{1}{2}\sum_{i, j} \left\langle [\mathrm{M}\varphi_i, \mathrm{N}\varphi_j], \eta\right\rangle [\varphi_i, \varphi_j],$$ where $\{\varphi_i\}_{i=1}^{n(n-1)/2}$ is any orthonormal basis for $\wedge^2T_p^*M$.

Now, we may compute directly from 