I have two vector fields and a diffeomorphism, G, on $\mathbb{R}^n$, where $n\geq2$: $$\mathbb{V}^i(x) = \frac{\partial{G^i}}{\partial{x^1}}(G^{-1}(x)),$$ $$\mathbb{W}^i(x) = \frac{\partial{G^i}}{\partial{x^2}}(G^{-1}(x)),$$ where $x = (x^1,...,x^n).$ I am asked to prove that $[\mathbb{V},\mathbb{W}] = 0,$ where, $$[\mathbb{X},\mathbb{Y}] = (\mathbb{X}\cdot\nabla)\mathbb{Y} - (\mathbb{Y}\cdot\nabla)\mathbb{X}.$$
So far I have found that: $$(\mathbb{W}\cdot\nabla)\mathbb{V}^i= \sum^{n}_{j=1} \frac{\partial{G^j}}{\partial{x^2}}(G^{-1}(x))\cdot\frac{\partial{G^{-1}}}{\partial{x^j}}(x)\cdot\frac{\partial^2{G^i}}{\partial{G^{-1}\partial{x^1}}}(G^{-1}(x)),$$ and: $$(\mathbb{V}\cdot\nabla)\mathbb{W}^i= \sum^{n}_{j=1} \frac{\partial{G^j}}{\partial{x^1}}(G^{-1}(x))\cdot\frac{\partial{G^{-1}}}{\partial{x^j}}(x)\cdot\frac{\partial^2{G^i}}{\partial{G^{-1}\partial{x^2}}}(G^{-1}(x)).$$ However, I don't see how these two can be equal and cancel to make $[\mathbb{V},\mathbb{W}] = 0.$ Is my use of the chain rule incorrect, or am I missing something?