Let $u:\mathbb{R}^d\rightarrow \mathbb{R}$ be a real-valued function on $\mathbb{R}^d$. Denote its Hessian by $\nabla^2 u = (\partial_{ij}u)$. I've encountered the real-valued expression $$- \sum_{i,j,l=1}^d\partial_{il}u\partial_{ij}u \partial_{lj}u, \tag{1}$$ and I am curious whether this quantity has a sign in general, in particular whether it's nonpositive.
If $\nabla^2 u$ is positive definite, then $$(1) = -\sum_{l=1}^d \langle{\nabla\partial_{l}u, (\nabla^2 u)\nabla\partial_{l}u}\rangle \le 0,$$ since each summand is $\le 0$. But for my purposes, I don't want to assume $\nabla^2 u$ is positive definite.