Let $\alpha:\mathbb R^n\longrightarrow \mathbb R,\,\, X:\mathbb R^n\longrightarrow \mathbb R^n,\,\,Y :\mathbb R^n\longrightarrow \mathbb R^n\,\,$ be three $\mathcal C^\infty$ maps.
We define for all vector $x\in\mathbb R^n$ $$[Y,X](x)=\dfrac{\partial Y}{\partial x}X(x)(x)-\dfrac{\partial X}{\partial x}Y(x)$$ where $\dfrac{\partial Z}{\partial x}(x)$ stands for the jacobian of the vectorial function $Z:\mathbb R^n\longrightarrow \mathbb R^p$
Now I have to find $U(x)$ and $V(x)$ in terms of $X(x), Y(x)$ and $\alpha(x)$ to satisfy the following
$$\color{red}{[\alpha Y,X](x)=\big(U(x)\nabla \alpha^T(x)\big)X(x)+V(x)[Y,X](x)}$$
Using ${[\alpha Y,X](x)=\big(U(x)\nabla \alpha^T(x)\big)X(x)+V(x)[Y,X](x)}$ we have $U(x)\nabla \alpha^T(x)= \big([\alpha Y,X](x)-V(x)[Y,X])X^{-1}(x)\big)$. But from here i get stuck. I don’t know how inverse the gradient operator.