 Solution

Write $q_{\alpha \beta }=g_{\alpha \beta }+n_{\alpha }{n}_{\beta }$ for the induced metric. By definition of the spatial covariant derivative,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 $n^{\mu }{n}_{\mu }=-1$. ThereforeThis verifies metric compatibility of the spatial covariant derivative directly, without assuming that the unprojected derivative of $q$ vanishes.