Let $\mathbb{P}(V)$ be a projective space of dimension $3$ and let $L_i$, $i=1,2,3$ be pairwise non-intersecting lines in $\mathbb{P}(V)$. If $\phi: L_1 \to L_1$ is a projective transformation, prove that there is a unique projective transformation $\Phi: \mathbb{P}(V) \to \mathbb{P}(V)$ such that $\Phi(L_i) = L_i$ and $\Phi_{|L_1} = \phi$.

It might be helpful to use that if $L_i = \mathbb{P}(U_i)$ then $U_i \oplus U_j = V$ and hence that there are unique maps $\alpha : L_1 \to L_2$ and $\beta: L_1 \to L_3$ such that $x$, $\alpha(x)$, $\beta(x)$ are collinear for all $x \in L_1$. (I was able to prove this, but cannot apply it properly).

Any help appreciated!


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