& b & c & a \\
 & 0 & d & 0 \\
 &  & 0 & 0 \\
 &  &  & 0
\end{array}
\right).$$ with the variables $x_1', x_2', x_3', x_4'$ changed to $x_1^{\prime\prime},x_2^{\prime\prime},x_4^{\prime\prime},x_3^{\prime\prime}$. This is the desired result.

\[proposition:trivial bundle\] A sum of two line bundles over a Bott manifold is trivial if and only if the total Chern class is trivial.

Let $B_n$ be a Bott manifold with the associated matrix $$\Lambda=\left(\begin{array}{cccc}
  0 & \c{2}{1} & \cdots & \c{n}{1} \\
    & 0 &        & \vdots  \\
    &   & \ddots & \c{n}{n-1}\\
    &   &   & 0
\end{array}
\right).$$ As before, let $x_j$ be the first Chern class of the line bundle $\gamma_j$ which is the pull-back bundle of the tautological line bundle of $P(\mathbb C\oplus \xi_{j-1})=B_j$ via the projection $B_n\to B_j$. Let $f_j = \sum_{i=1}^{j-1}\c{j}{i}x_i$. For an element $\alpha \in H^2(B_n)$, let $\gamma^{\alpha}$ be the complex line bundle over $B_n$ with $c_1(\gamma^\alpha) = \alpha$. Let $\xi =
\gamma^{\alpha} \oplus \gamma^{\beta}$ be the sum of two line bundles such that $c(\xi)=1$, and $\alpha = \sum_{j=1}^n a_j x_j$ and $\beta = \sum_{j=1}^{n} b_j x_j$. Then $$\begin{aligned}
1 = c(\xi) &= c(\gamma^\alpha) c(\gamma^\beta) \\
        &= (1+ \alpha)(1+\beta) \\
        &= 1 