= Solution
Contracting the assumed <Riemann curvature tensor> gives the <Ricci tensor>
$$
R_{bd}=g^{ac}R_{abcd}=\frac{R}{12}(4g_{bd}-g_{bd})=\frac R4g_{bd}.
$$
The <contracted Bianchi identity> and <metric compatibility> now imply
$$
\frac14\nabla_bR=\nabla^aR_{ab}=\frac12\nabla_bR.
$$
Thus \b[$\nabla_bR=0$, so the scalar curvature is constant on each connected component of spacetime.] The sectional curvature is consequently the constant $\kappa=R/12$. This is the four-dimensional case of the <Schur theorem in pseudo-Riemannian geometry>.
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