proof of $d\alpha (X,Y)=X(\alpha (Y))$ $$ $Y(\alpha (X))$ $$ $\alpha ([X,Y])$ (local coordinates)
Since this result is local (in other words, the identity^{} holds on the whole manifold if and only if its restriction^{} to every coordinate patch of the manifold holds), it suffices to demonstrate it in a local coordinate system. To do this, we shall compute coordinate expressions for the various terms and verify that the sum of terms on the righthand side equals the lefthand side.

$$d\alpha (X,Y)=({\alpha}_{j,i}{\alpha}_{i,j}){X}^{i}{Y}^{j}={\alpha}_{j,i}{X}^{i}{Y}^{j}{\alpha}_{i,j}{X}^{i}{Y}^{j}$$ 


$$X(\alpha (Y))={X}^{i}{\partial}_{i}({\alpha}_{j}{Y}^{j})={X}^{i}{\alpha}_{j,i}{Y}^{j}+{X}^{i}{\alpha}_{j}Y^{j}{}_{,i}$$ 


$$Y(\alpha (X))={Y}^{j}{\partial}_{j}({\alpha}_{i}{X}^{i})={Y}^{j}{\alpha}_{i,j}{X}^{i}+{Y}^{j}{\alpha}_{i}X^{i}{}_{,j}$$ 


$$\alpha ([X,Y])={\alpha}_{i}({X}^{j}Y^{i}{}_{,j}{Y}^{j}X^{i}{}_{,j})={\alpha}_{i}{X}^{j}Y^{i}{}_{,j}{\alpha}_{i}{Y}^{j}X^{i}{}_{,j}$$ 

Upon combining the righthand sides of the last three equations and cancelling common terms, we obtain

$${X}^{i}{\alpha}_{j,i}{Y}^{j}+{X}^{i}{\alpha}_{j}Y^{j}{}_{,i}{Y}^{j}{\alpha}_{i,j}{X}^{i}{\alpha}_{i}{X}^{j}Y^{i}{}_{,j}$$ 

Upon renaming dummy indices (switching $i$ with $j$), the second and fourth terms cancel. What remains is exactly the righthand side of the first equation. Hence, we have

$$d\alpha (X,Y)=X(\alpha (Y))Y(\alpha (X))\alpha ([X,Y])$$ 
