I am working on a (Lorentzian) manifold $(M,g)$, and I know that in local coordinates $g$ takes the form \begin{equation} \label{1} g = h_{ij}\mathrm{d}x^i \mathrm{d}x^j + k_{\mu \nu} \mathrm{d} x^\mu \mathrm{d} x^\nu, \end{equation} where, say, the indices $i,j$ run from $0$ to $1$ and $\mu, \nu$ run from $2$ to $3$. I want to say that, in a suitable sense, the Levi-Civita connection of $g$ decomposes into a direct sum of Levi-Civita connections of $h$ and $k$, $$\nabla^{(g)} = \nabla^{(h)} \oplus \nabla^{(k)}.$$ I have a feeling that this is trivial, but got into a slight mess trying to prove it. My intuition is that the above form of the metric $g$ means that the tangent bundle $TM$ must be a sum of the form $TM = TH \oplus TK$. Any vector field $X \in \Gamma(TM)$ then decomposes into a sum $X = X^{(h)} \oplus X^{(k)}$, where $X^{(h)} \in \Gamma(TH)$, $X^{(k)} \in \Gamma(TK)$, and the only sensible definition of a direct of sum of connections I could come up with was $$ (\nabla^{(h)}\oplus \nabla^{(k)} )(X, Y) := (\nabla^{(h)}(X^{(h)}, Y^{(h)})) \oplus (\nabla^{(k)}(X^{(k)}, Y^{(k)})).$$ However, this doesn't seem to (although I feel like I am wrong) define a connection on $TM$ (I couldn't convince myself that the Leibniz rule holds).
I have a strong feeling that such a decomposition of $\nabla^{(g)}$ must hold, however. For one, I have calculated the Christoffel symbols of $g$ and it turns out that all the ones that mix Roman and Greek indices vanish. I feel like I am missing something obvious.
P.S. I do not a priori know that $M= H \times K$. Does the form of the metric imply this?
