I know that there is two convenient for exterior derivative:
$$d\omega (X,Y)=\frac{1}{2}(X\omega(Y)-Y\omega(X)-\omega([X,Y])),$$ and $$d\omega (X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y]),$$ Maple uses the second version. Does there exist various version of Lie derivative or Lie bracket such as exterior derivative?