Additional properties are consistent with that of the Lie bracket.

The Lie bracket of these derivations is given as follows.

The Lie bracket on this space is given by the commutator:

One also has where is the usual Lie bracket, which follows from the Jacobi identity.

If the basis is holonomic then the Lie brackets vanish, .

This can be used to define the Lie bracket of vector fields as follows.

Furthermore, there is a "product rule" for Lie brackets.

The geometrical interpretation of the Lie bracket can be applied to the last of these equations.

This failure of closure under Lie bracket is measured by the curvature.

The Lie bracket of is given by the commutator of matrices.