Solution
= Solution
Let $D$ be the <Levi-Civita connection> of $h$. Apply $(D_cD_d-D_dD_c)$ to a tangent vector, replace each $D$ by the tangential projection of $\nabla$, and decompose $\nabla n$ using $K_{ab}$. Comparison with the intrinsic <Riemann curvature tensor> gives the <Gauss equation>
$$
\boxed{{}^{(3)}R_{abcd}=h_a{}^eh_b{}^fh_c{}^gh_d{}^h\,{}^{(4)}R_{efgh}-K_{ac}K_{bd}+K_{ad}K_{bc}}.
$$
Changing the sign convention for $K_{ab}$ does not alter these quadratic terms.