= Solution
The <product Riemannian metric> has block matrix
$$
g=\begin{pmatrix}g_M(x)&0\\0&g_N(y)\end{pmatrix},\qquad g^{-1}=\begin{pmatrix}g_M(x)^{-1}&0\\0&g_N(y)^{-1}\end{pmatrix}.
$$
In the given <Christoffel symbols> formula, every mixed coefficient vanishes. Off-diagonal metric entries are zero, $\partial_{y^\alpha}g_{ij}=0$, and $\partial_{x^i}g_{\alpha\beta}=0$. These facts give $\Gamma^A_{i\alpha}=0$ for every output index $A$, as well as $\Gamma^\alpha_{ij}=0$ and $\Gamma^i_{\alpha\beta}=0$. The remaining pure blocks are precisely the Christoffel symbols of the respective factor metrics.
For the lifted fields of (i), $\partial_{x^i}Y^\alpha=0$. Consequently
$$
\nabla_XY=\sum_{i,\alpha}X^i\partial_{x^i}Y^\alpha\,\partial_{y^\alpha}+\sum_{A,i,\alpha}\Gamma^A_{i\alpha}X^iY^\alpha\partial_A=0.
$$
Thus $\boxed{\nabla_XY=0}$. The same calculation gives $\nabla_YX=0$; differentiating two fields from one factor gives the lifted factor covariant derivative. This identifies the product <Levi-Civita connection> with the <product affine connection> and proves the curvature splitting used in the root solution.
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