**Statement.** Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$. It is *well known* that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a *projective variety* to the projectivisation $\mathbb PV$. 

**Question.** I would like to find a precise reference to this statement, is there such a reference? 

(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the **statement** holds. 
But I can't find any place in the literature where this **statement** about projective bundles is stated and I need to find such a reference :( )

**PS.** I realised, that Section 6.2 of the beautiful paper by Arnaud Beauville 
https://arxiv.org/abs/1507.02476
would do as a reference to the **statement**. Yet this section speaks of something far more general. I still hope for something more direct, maybe along the lines of what Angelo suggested in his comment (or at least explicitly stated).