Levi Civita Connection of the product manifold with product metric

Question

I’m trying to prove exercise 1a) down here that I took from Do Carmo "Riemannian Geomtetry", chapter 6, but I’ve been stuck into it for a while. Particularly, if $X_{1} \in \Gamma(TM_{1}), X_{2} \in \Gamma(TM_{2})$ I guess we're considering them as $X_{1} \oplus 0$, $0 \oplus X_{2} $ as $T(M_{1} \times M_{2}) = TM_{1} \oplus TM_{2}$ and their coefficients live in $C^{\infty}(M_{1} \times M_{2})$. First, I tried to write Christoffel's symbols of both connections in the two sides of the equations, but they aren't the same because there are some mixed derivatives of the coefficients that seem to survive. Then,I tried to prove that $\nabla^{1} + \nabla^{2}$ is the riemannian connection of $M_{1} \times M_{2}$ by saying it's well defined, torsion free and compatible with che product metric, but I had some troubles especially proving it's torsion free, because I think you can't actually say that $[X_{1}+ X_{2},Y_{1}+Y_{2}]=[X_{1} \oplus 0,Y_{1} \oplus 0] +[0 \oplus X_{2},0 \oplus Y_{2}].$ Where am I doing wrong? Can someone help me, please?