$\alpha_1$ has exactly two non-zero coefficients. Then (1) and (2) imply that so is $\alpha_2$, and $\beta_1$ and $\beta_2$ are either zero or have exactly two non-zero coefficients at the same places as $\alpha_1$. This means that the bundles $\xi_1$ and $\xi_2$ are pullbacks of bundles over $(\mathbb CP^1)^2$. Hence those bundles are in stable range and hence they are classified by their Chern classes. Thus $\xi_1$ and $\xi_2$ are isomorphic.

The case when $\alpha_1$ has only one non-zero coefficient can be proved similarly.

Twist number and Cohomological complexity {#section: twist number}
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The [*twist number*]{} of a Bott tower $\{B_j \mid j=0,\ldots,n\}$ is the number of nontrivial fibrations $B_j\to B_{j-1}$ in the sequence. However there may be several Bott tower structures for a Bott manifold, so the twist number may not be well-defined for Bott manifolds. In this section we show that the twist number of a Bott manifold is well-defined, namely we show that the twist numbers of any Bott tower structure of a Bott manifold is constant.

For an $n$-stage Bott manifold $M$ its cohomology ring is isomorphic to $$H^\ast(M) \cong {\mathbb{Z}}[x_1, \ldots, x_n]/I,$$ where $I=<x_j(x_j - f_j) : j=1, \ldots, n>$ and $f_j = \sum_{i=1}^{j-1} \c{j}{i}x_i$ 