= Solution
Using the normalization of the timelike <unit normal>, the <spatial projection tensor> satisfies
$$
P^\mu{}_\alpha n_\mu=n_\alpha+(n^\mu n_\mu)n_\alpha=0.
$$
Likewise $P^\mu{}_\alpha n^\alpha=0$ and $P^\mu{}_\alpha P^\alpha{}_\nu=P^\mu{}_\nu$. The <spacetime> form of the <induced metric> is therefore
$$
h_{\alpha\beta}=P^\mu{}_\alpha P^\nu{}_\beta g_{\mu\nu}=g_{\alpha\beta}+n_\alpha n_\beta.
$$
Its restriction to the slice is the given three-metric. <Metric compatibility> of the four-dimensional <Levi-Civita connection> gives
$$
\nabla_\ell h_{\alpha\beta}=(\nabla_\ell n_\alpha)n_\beta+n_\alpha\nabla_\ell n_\beta.
$$
Apply the three projectors defining the <spatial covariant derivative>. In the first summand $P^\beta{}_j n_\beta=0$, and in the second $P^\alpha{}_i n_\alpha=0$. Both summands vanish, so
$$
\boxed{P^\mu{}_\alpha n_\mu=0,\qquad h_{ij|k}=0.}
$$
This proves the <metric compatibility of the spatial covariant derivative> without treating the <spatial projection tensor> itself as covariantly constant.
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