have in constructing the expansion (\[mainidea\]) with this criteria. This freedom and the non-uniqueness of the construction above is not a problem: the operators $\cL_{(p,q)}$ and the corresponding vectors (see below) that we are going to define should be though of as a “basis" for the construction in terms of Lie derivative of each of the terms in the expansion (\[Taylorfamily\]). This “basis" is not unique, but the result at each order is.

On the other hand, it is straightforward to check that these operators are linear and satisfy the Leibnitz rule and hence they are derivatives. We can also check that they satisfy the rest of conditions of the theorem stated in \[sebastiano\]. Therefore, for each of them there is a vector field $\xi_{(p,q)}$ such that \_[\_[(p,q)]{}]{}T := \_[(p,q)]{}T      (p,q). \[lieder\]

In the particular case when $\Phi$ is a group of diffeomorphisms we recover the previous case (subsection \[groupdif\]), and $\cL_{(p,q)}=0$ if $p+q>1$.

Using the differential operators we have just introduced we can express the Taylor expansion (\[Taylorfamily\]) of $\Phi_{\lambda,\Omega}$, up to fourth order in $\lambda$ and $\Omega$, in terms of the Lie derivatives associated with the vector fields $\xi_{(p,q)}$ (\[lieder\]): \^\_[,]{}T & & = T + \_[\_[(1,0)]{}]{}T + 