Abelian. On the other hand, it allow us to make the following useful decomposition of $\phi_{\lambda,\Omega}$ into two one–parameter groups of diffeomorphisms (flows) that remain implicitly defined by the equalities \_[,]{}=\_[,0]{}\_[0,]{} =\_[0,]{}\_[,0]{}. \[philambda\] The action of the flows $\phi_{\lambda,0}$ and $\phi_{0,\Omega}$ is generated by two vector fields, $\eta$ and $\zeta$ respectively, acting on the tangent space of $\cM\times\RR^2$. The Lie derivatives of a generic tensor $T$ with respect to $\eta$ and $\zeta$ are \_T&: = \_[0]{} (\^\*\_[,0]{}T-T) = &\_[=0]{} =\_,\
\_T&:= \_[0]{} (\^\*\_[0,]{}T-T) = &\_[=0]{} =\_, where the superscript ${}^*$ denotes the pull–back map associated with the corresponding diffeomorphism [@BMMS]. Because the group is Abelian the vector fields $\eta$ and $\zeta$ must commute =0. The Taylor expansion of the pull–backs $\phi^*_{\lambda,0} T,\phi^*_{0,\Omega} T$ is given by (see [@BMMS]) \^\*\_[,0]{}T&=\_[k=0]{}\^ \_[=0]{}= &\_[k=0]{}\^\^k\_T ,\
\^\*\_[0,]{}T&=\_[k=0]{}\^ \_[=0]{}= &\_[k=0]{}\^ \^k\_T. From this, using (\[philambda\]), we can derive the Taylor expansion of the two–parameter group of pull–backs $\phi^*_{\lambda,\Omega}T$: \^\*\_[,]{}T = \_[k,k’=0]{}\^ \_ = \_[k,k’=0]{}\^ \^k\_\^[k’]{}\_T.\[Taylorgroup\]

Two–parameter families of diffeomorphisms {#familydiffeo}
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Let us now consider the general case of a two–parameter family of diffeomorphisms $\Phi$:

[cccc]{} :&\^2 & & \^2\
&(p,,)&|& (\_[,]{}(p),,) .\

In this generic case the diffeomorphisms $\Phi_{\lambda,\Omega}$ do not form in general 