We check various properties of these invariant bases of vector fields and forms. First note that the structure constants for the right invariant vector fields are the negatives of those for the left invariant vector fields.
The Lie derivatives of XL and OmegaL with respect to XR vanish:
Let us check explicitly the left invariance of the vector field XL[4].
Define points and and compute .
Evaluate at and at .
Pushforward by . Since Y the vector field X is left invariant.
Alternatively, we can verify the left invariance of using the second calling sequence for Pushforward to see that is unchanged.
The left invariance of the form OmegaL[2] is similarly verified (by observing that