Sectional curvatures from Ricci curvature in dimension three ($K_{ij}=\frac{1}{2}({\mathrm{Ric}}_{ii}+{\mathrm{Ric}}_{jj}-{\mathrm{Ric}}_{kk})$)

For an orthonormal triple and distinct $i,j,k$, tracing sectional curvature gives ${\mathrm{Ric}}_{ii}=K_{ij}+K_{ik}$. Solving the three equations gives the displayed formula. Equivalently $K_{ij}={\mathrm{Ric}}_{ii}+{\mathrm{Ric}}_{jj}-\mathrm{Scal}/2$. This pointwise assertion does not require the triple to diagonalize the Ricci curvature.