= Solution
Use product coordinates $x^i$ on $M$ and $y^\alpha$ on $N$. The lifted <vector fields> have the form $X=\sum_iX^i(x)\partial_{x^i}$ and $Y=\sum_\alpha Y^\alpha(y)\partial_{y^\alpha}$. The coefficients of each depend only on its own coordinate set, and the coordinate partial derivatives commute. The <Lie bracket of vector fields> formula therefore gives
$$
[X,Y]=\sum_{i,\alpha}X^i\partial_{x^i}Y^\alpha\,\partial_{y^\alpha}-Y^\alpha\partial_{y^\alpha}X^i\,\partial_{x^i}=0.
$$
Thus $\boxed{[X,Y]=0}$ for fields lifted separately from the two factors. The lift hypothesis matters: arbitrary product-manifold fields can have coefficients depending on both factors.
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