Corollary of Leibniz rule for Lie bracket
This is an identity that uses the Leibniz rule to measure the failure of the Lie bracket operation from being -linear.
We prove this by showing that for any , both sides evaluate to the same thing. Let's do this. Simplifying the right side yields:
Applying the Leibniz rule for on the product of functions and , this simplifies to:
which is precisely equal to the left side.
We prove this as well by taking any test function . Simplifying the right side yields:
which is the same as the left side.