# Corollary of Leibniz rule for Lie bracket

From Diffgeom

## Statement

This is an identity that uses the Leibniz rule to measure the failure of the Lie bracket operation from being -linear.

Let be smooth vector fields on a differential manifold and be in . We then have:

## Applications

## Proof

### First identity

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.

### Second identity

We prove this as well by taking any test function . Simplifying the right side yields:

which is the same as the left side.