Two Kinds of Curvature Operators in Dimension Four

We prove an invariant identity relating the curvature operators of the first and second kind in dimension four. Under the natural identification $S_0^2\cong {\mathrm{\Lambda }}^{+}\otimes {\mathrm{\Lambda }}^{-}$, the trace-free Ricci tensor contributes through a single commutator term. As a consequence, we relate the curvature operator of the second kind on products of unit self-dual and anti-self-dual two-forms to biorthogonal sectional curvature. We then classify closed oriented four-manifolds satisfying $K_{\max }^{}\le S\mathrm{/}6$, where $K_{\max }^{}$ denotes the maximum biorthogonal sectional curvature and $S$ is the scalar curvature. The classification includes the cases in which the scalar curvature vanishes.