I$.\
Indeed, since, by assumption, $b_{k}$ is non-singular, there exists $u\in V_{k}$ such that $b_{k}(u,u_j)\neq 0$. Let $c_{k}\in
A_{k}$ be the element $$c_{k} = ((t'_1,\dots,t'_n),(0,\dots ,0)),$$ with $t'_k=(u'_{1}, \dots , u'_{m_{k}})$ such that $u'_{j}=u$ and $u'_{l} =0$ for all $l\neq j$. Then, $$\begin{aligned}
\lefteqn{\lambda_{c_k}((t_1,\dots ,t_n),(s_1,\dots ,s_n))}\\
&=&((t'_1,\dots,t'_n),(0,\dots,0))((t_1,\dots,t_n),(s_1,\dots,s_n))\\
&&-((t'_1,\dots,t'_n),(0,\dots,0))\\
&=&((t'_1+t_1,\dots,t'_n+t_n),(s_1,\dots,s_n))-((t'_1,\dots,t'_n),(0,\dots,0))\\
&=&((t_1,\dots,t_n),(s_1,\dots,s_n)-b((t'_1,\dots,t'_n),(t_1,\dots,t_n)))\\
&=&((t_1,\dots,t_n),(s_1-b'_1(t'_1,t_1),\dots,s_n-b'_n(t'_n,t_n)))\in
I\cap A_k,
 \end{aligned}$$ with $$s_k-b'_k(t'_k,t_k)
=s_k -b_{k}(u,u_{j})
  e_{r_{k}}^{(k)},$$ (recall that $(t_z,s_z-b'_z(t'_z,t_z)) =0$ for all $z\neq k$). Now, $$\begin{aligned}
\lefteqn{((t_1,\dots ,t_n),(s_1,\dots ,s_n))}\\
&&-((t_1,\dots
,t_n),(s_1-b'_1(t'_1,t_1),\dots,s_n-b'_n(t'_n,t_n)))\\
&=&((0,\dots ,0),(b'_1(t'_1,t_1),\dots ,b'_n(t'_n,t_n)))\in I\cap
A_k,\end{aligned}$$ with $b'_1(t'_k,t_k)= b_{k}(u,u_{j}) e_{r_{k}}^{(k)}$. Thus $a_{k,r_k}\in I$ and the claim follows.

[*Claim 2*]{}: If $u_j\neq 0$ for some $j$ with $1\leq j < r_{k}$, then $a_{k,j} +a_{k,r_{k}}\in I$.\
As in the proof of Claim 1, we see that $$\begin{aligned}
\lefteqn{((t_1,\dots ,t_n),(s_1,\dots
,s_n))-\lambda_{c_k}((t_1,\dots
,t_n),(s_1,\dots ,s_n))}\\
&=&((0,\dots ,0),(b'_1(t'_1,t_1),\dots ,b'_n(t'_n,t_n)))\in I\cap
A_k,\end{aligned}$$ with $b'_k(t'_k,t_k)= b_{k}(u,u_{j})(e_{j}^{(k)}
+e_{r_{k}}^{(k)})$ and $b_{k}(u,u_{j})\neq 0$. Thus $a_{k,j}+a_{k,r_k}\in I$ and the claim follows.

[*Claim 3*]{}: If $\mu_{r_k}=0$ and $\mu_{i}\neq 0$ for some $i$ with $1\leq i <r_{k}$ then $c_{k+1}\in I$ for some $c_{k+1}\in
A_{k+1}$ such that $c_{k+1} =((t'_1,\dots,
t'_n),(s'_1,\dots,s'_n))$, $t'_{k+1}=(v_{1}, \dots , v_{m_{k}+1})$, and some $v_{l}\neq 0$.\
Indeed, since $f_{k+1}$ has order $p_k$ and $\mu_i\neq 0$, there exists $v\in V_{k+1}$ such that $f_{k+1}^{\mu_{i}}(v)\neq
 v$. Let $$c'_{k+1} =((t''_1,\dots ,t''_n),(0,\dots,0))\in A_{k+1}$$ be the 