Two questions from the proof of Harnack's Inequality

Question

1. My first question concerns the first line on page 8 of the following: https://web.math.princeton.edu/~const/maxhar.pdf. Explicitly, it is the following estimate: $$\sum_{i,j}|a^{ij}_{kl}v_iv_l|+|a^{ij}_k(v_iv_j)_l|\leq\epsilon|\nabla^2v|^2+C|\nabla v|^2.$$ where all the subscripts denote take partial derivatives of, and $a^{ij}$ forms a symmetric, positive definitely matrix pointwise. On the RHS, $C$ is a constant depending on $\epsilon$. I'm assuming the idea is that the first term on the LHS is bounded by the $|\nabla v|^2$ term, and the second term on the LHS by the $|\nabla^2 v|^2$ term, but I'd appreciate if anyone could be more explicit.
2. My second question is again on page 8, this time near the bottom, when they use uniform ellipticity to deduce $$\sum_{i,j,k,l}a^{ij}a^{kl}v_{ik}v_{jl}\geq\gamma^2|\nabla^2v|^2.$$ I'm having trouble seeing this since the indices are intertwined on the LHS.

Answer 1

Asked a professor.

1. We have by Cauchy's estimate with $\epsilon$ on the second term on the RHS $$\sum_{i,j} a^{ij}_k (v_iv_j)_l\leq C\sum_{i,j}v_{il}v_j\leq \epsilon |D^2v|^2+C |Dv|^2.$$ The first term is just regular Cauchy-Schwarz.
2. We can diagonalize the coefficient matrices $a^{ij}$ and $a^{kl}$ at a point and then use the ellipticity.