$z,z'\in \zz$ and $w\in \fA^1$ we have $$\begin{aligned}
 \label{equ:gdzz'}
    \exp(\gd_z)(z')=&\big(\exp_\circ(z)\circ z'\big)\exp_*(z),\\
   \exp(\gd_z)(w)=&(\exp_*(z))^{-1}\big(\exp_*(z)* w\big).\end{aligned}$$

\[prop:Phi\] For $z\in \zz$ define $\Phi_z:\fA^1\to \fA^1$ by $$\label{equ:Phi}
    \Phi_z(w):=(1-z)\left(\big(1-z\big)^{-1}*w\right)\qquad (w\in
    \fA^1).$$ Then $\Phi_z$ is an automorphism of $\fA^1$ and we have $$\label{equ:Phiexp}
 \Phi_z(w)=\exp(\gd_t)(w), \qquad\text{where }t=\log_\circ(1+z)\in
 \zz.$$ All the $\Phi_z$ commute. Moreover, after restricting the derivation $\gd_t$ to $\fA^0$ we can regard $\Phi_z$ as an automorphism of $\fA^0$, If we extend the derivation $\gd_t$ to the whole $\fA$ as in Cor. \[cor:gdz\] then we can regard $\Phi_z$ as an automorphism of $\fA$.

The key point is that $\gd_t$ sends $\fA^0$ to $\fA^0$ as a derivation by Cor. \[cor:gdz\]. Hence $\exp(\gd_t)$ is an automorphism on $\fA^1$ as well as on $\fA^0$.

The next three results are generalizations of Prop. 7, its corollary, and Prop. 8 of [@IKZ], respectively. The proofs there can be easily adapted into our situation because the $\sha$-product is essentially the same (note that the the only essentially new phenomenon in the higher level MPV cases is that there are exponent shiftings on the roots of unity in our stuffle product.)

\[prop:d\] Define the map $d:\fA\to \fA$ by $d(w)=b_0\sha w-b_0 w$. Then $d$ is a derivation and by setting $u$ as a formal parameter we 