= Solution
Write $q_{\alpha\beta}=g_{\alpha\beta}+n_\alpha n_\beta$ for the <induced metric>. By definition of the <spatial covariant derivative>,
$$
D_\mu q_{\alpha\beta}
=P_\mu{}^\rho P_\alpha{}^\sigma P_\beta{}^\lambda
\nabla_\rho(g_{\sigma\lambda}+n_\sigma n_\lambda).
$$
The spacetime <Levi-Civita connection> is metric compatible, so $\nabla g=0$. The remaining terms contain either $P_\alpha{}^\sigma n_\sigma$ or $P_\beta{}^\lambda n_\lambda$, each zero because $n^\mu n_\mu=-1$. Therefore
$$
\boxed{D_\mu q_{\alpha\beta}=0.}
$$
This verifies <metric compatibility of the spatial covariant derivative> directly, without assuming that the unprojected derivative of $q$ vanishes.
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