  We establish a new algebraic characterization of sectional curvature bounds
$\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock
formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the
Kulkarni-Nomizu product, we provide a simple formula for such curvature terms.
We also give an application of the Bochner technique to closed $4$-manifolds
with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new
insights into the Hopf Conjecture, without any symmetry assumptions.
