Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 19 / 5 / i / Solution

Use product coordinates $x^i$ on $M$ and $y^{\alpha }$ on $N$. The lifted vector fields have the form $X={\sum }_i{X}^i(x){\partial }_{x^i}$ 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 givesThus $\boxed{[X,Y]=0}$ for fields lifted separately from the two factors. The lift hypothesis matters: arbitrary product-manifold fields can have coefficients depending on both factors.