It's obvious that the Lie derivative defined in terms of Lie brackets is antisymmetric. But what is an intuitive way to visualize the antisymmetry in the 'differentiating along a flow' definition?
Thanks!
It's obvious that the Lie derivative defined in terms of Lie brackets is antisymmetric. But what is an intuitive way to visualize the antisymmetry in the 'differentiating along a flow' definition? Thanks! 


Whichever definition you use that involves differentiating along a flow, it ought to have the property that $L_X X = 0$; in other words, for reasons that should be clear, the Lie derivative is alternating. In the presence of bilinearity, this is equivalent to antisymmetry. 

