 Solution

Let $D$ be the Levi-Civita connection of $h$. Apply $(D_c{D}_d-D_d{D}_c)$ 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 equationChanging the sign convention for $K_{ab}$ does not alter these quadratic terms.