I understand that one must use the following tensor maximum principle in order to prove that positive Ricci curvature is preserved under the Ricci flow:
$\textbf{The Tensor Maximum Principle.}$ Let $g(t)$ be a smooth $1$-parameter family of Riemannian metrics on a closed manifold $\mathcal{M}^n$. Let $\alpha(t) \in \mathcal{C}^{\infty}(T^{*}\mathcal{M}^n \otimes_{S} T^{*} \mathcal{M}^n)$ be a symmetric $(2, 0)$ tensor satisfying the semilinear heat equation: $$\frac{\partial}{\partial t}\alpha \geq \Delta_{g(t)} \alpha + \beta$$ where $\beta(\alpha, g, t)$ is a symmetric $(2, 0)$ tensor which is locally Lipschitiz in all its arguments and satisfies the $\textbf{null eigenvector assumption}$ that $$\beta(V, V)(x, t) = (\beta_{ij} V^{i} V^{j})(x, t) \geq 0$$ whenever $V(x, t)$ is a null eigenvector of $\alpha(t)$, that is, whenever $$(\alpha_{ij} V^{j})(x, t) = 0$$ If $\alpha(0) \geq 0$ (that is, if $\alpha(0)$ is positive semidefinite), then $\alpha(t) \geq 0$ for all $t \geq 0$ such that the solution exists.
Now, on the books I'm reading, the proof of that fact (the preservation of positive Ricci curvature under the Ricci flow, not the tensor maximum principle) goes like this:
Since $$\frac{\partial}{\partial t} R_{jk} = \Delta R_{jk} -Q_{jk}$$ where $$ Q_{jk} = -3 R R_{j k}+6 g^{p q} R_{j p} R_{q k}-\left(2|\mathrm{Ric}|^{2}-R^{2}\right) g_{j k}$$ And since at any point and time where the Ricci tensor has a null eigenvector, it has at most two non-zero eigenvalues, whence $|\text{Rc}|^2 \geq \frac{R^2}{2}$, the desired result follows immediately from the tensor maximum principle.
The thing is, I'm having trouble seeing why a particular hypothesis in the tensor maximum principle holds true here (namely, the locally Lipschitz one - the null eigenvector assumption is clearly satisfied). Of course, $\text{Ric}_{g(t)}$ must play the role of $\alpha$ and $Q$ the one of $\beta$.
But why is it that $Q$ is locally Lipschitz?
I assume that given a $(2, 0)$ tensor $T$, then $Q(T)$ would be given by $$(Q(T))_{jk}= - 3\text{tr}(T)T_{jk} + 6 g^{pq}T_{jp}T_{qk} - 2(|T|^2 - \text{tr}(T)^2)g_{jk}$$
I found this pretty hard to answer, but all the books I've read don't even address it so it must probably be something obvious... however I can't see how on Earth this is true. I would greatly appreciate it if someone clarified this issue for me.
Thanks in advance!