The composition of Lie transforms can be written as products of
exponentials of Lie derivative operators. In general, Lie derivative
operators do not commute.
If *A* and *B* are non-commuting operators, then the exponents do not
combine in the usual way:

So it will be helpful to recall some results about exponentials of non-commuting operators.

The commutator is bilinear and satisfies the Jacobi identity

which is true for all *A*, *B*, and *C*.

We introduce a notation _{A} for the commutator with respect to
the operator *A*:

In terms of the Jacobi identity is

We can check this term by term.

using *e*^{-C} *e*^{C} = *I*, the identity operator.
Using the same trick, we find

More generally, if *f* can be represented as a power series then

For instance, applying this to the exponential function yields

Using equation (5.467), we can rewrite this as

**Exercise 5.30.** **Commutators of Lie derivatives**

**a**. Let *W* and *W*' be two phase-space state functions.
Use the Poisson-bracket Jacobi identity (3.92)
to show

**b**. Consider the phase-space state functions that give the
components of the angular momentum in terms of rectangular canonical
coordinates

**c**. Relate the Jacobi identity for operators to the
Poisson-bracket Jacobi identity.

**Exercise 5.31.** **Baker-Campbell-Hausdorff**

Derive the rule for combining exponentials of non-commuting operators: