 Static Killing field is a Ricci eigenvector

A timelike Killing vector with hypersurface orthogonality admits local coordinates with $k={\partial }_t$ and $d{s}^2=N(\mathbf{x}{)}^{2}d{t}^2-h_{ij}(\mathbf{x})d{x}^{i}d{x}^j$. The only connection coefficients with time indices that can be nonzero are ${\Gamma }_{0i}^0={\partial }_i\log N$ and ${\Gamma }_{00}^i=N{h}^{ij}{\partial }_{j}N$. Substitution into the Ricci tensor gives $R_{0i}=0$. Thus $R_{ab}{k}^b$ has only its time component and is proportional to $k_a$. Equivalently $k_d{R}^d{}_{[a}{k}_{b]}=0$. This local statement does not require the Vacuum Einstein equations and is independent of the overall curvature-sign convention.