+ (\alpha + \beta) + \alpha\beta.\end{aligned}$$ Therefore $\alpha + \beta = 0$ and $\alpha \beta = 0$, which implies $\alpha^2=0$ in $H^\ast(B_n)$. On the other hand, $$\begin{aligned}
    \alpha^2 = 0 &\Leftrightarrow \sum_{j=1}^{n}(a_jx_j)^2 +
    \sum_{1\leq i<j \leq n} 2a_ja_i x_j x_i = \sum_{j=1}^n a_j^2 (
    x_j^2 - f_jx_j) \nonumber\\
    & \Leftrightarrow  a_j^2 \c{j}{i} = - 2a_ja_i \text{ for all }
    i<j \label{equation:trivial chern class}\end{aligned}$$ Thus $\xi = \gamma^{\alpha} \oplus \gamma^{-\alpha}$ with $\alpha = \sum_{j=1}^n a_j x_j \in H^2(B_n)$ and $a_j^2 \c{j}{i} = - 2a_j a_i$ for all $ 1 \leq i < j \leq n$.

Now, we prove the proposition by induction on $n$. If $n=2$, then the dimension of $\xi$ is equal to the dimension of $B_n$, so we are in the stable range. Hence the total Chern class classifies the complex vector bundle, so the proposition is true for $n=2$. Assume the lemma is true for $B_{n-1}$. We now prove the lemma for $B_n$. These are three cases to consider

**Case 1** $a_n = 0$.

In this case, $\xi =\pi_n^\ast(\eta)$, where $\eta =\gamma^{\alpha}\oplus \gamma^{-{\alpha}}$ over $B_{n-1}$. By the assumption, $c(\eta)=1$, and by the induction hypothesis, $\eta$ is trivial. So is $\xi$.

**Case 2** $a_n \neq 0$ and $a_k=0$ for 