Z[x_1, x_2, x_3, x_4]/I$ where $I$ is the ideal generated by $$x_1^2,\  x_2(x_2-ax_1),\  x_3(x_3 - bx_1), \ x_4(x_4 - dx_3 - cx_1).$$ We apply Lemma \[lemma:2-1\] to $B_3$ whose associated matrix is $$B=
\left(
\begin{array}{c c c}
0 & a & b \\
 & 0 & 0 \\
 &  & 0 \\
\end{array}
\right),$$ which results exchanging the second and third columns of $B$. The effect of the above procedure also exchanges the second and third stages of the Bott tower of $B_4$, and as a result the variable $x_2$ and $x_3$ will be exchanged. Namely, the variables $x_1, x_2, x_3, x_4$ will be changed to $x_1', x_3', x_2', x_4'$. Therefore, with the changed variables $x_1', x_2', x_3', x_4'$, $B_4$ is diffeomorphic to $B_4'$ with the assocoated matrix $$A'=
\left(
\begin{array}{c c c c}
0 & b & a & c \\
 & 0 & 0 & d \\
 &   & 0 & 0 \\
 &  &  & 0
\end{array}
\right).$$ We now apply Lemma \[lemma:2-1\] to $A'$. This means that we are exchanging the third and fourth stages of the Bott tower of $B_4'$ to get $B_4 ^{\prime\prime}$ with the associated matrix $$A^{\prime\prime}=
\left(
\begin{array}{c c c c}
0 