Pullback bundle of bundle from real projective space to complex projective space

Question

Let $V$ be complex vector space. As we know there is canonical map $\mathbb P(V_\mathbb{R})\rightarrow\mathbb P(V)$. Furthermore, let $H(V)=\{(\ell,v) : v \in\ell\}$ and we have a bundle $H(V)\rightarrow\mathbb P(V)$.

How can I see that realification pullback of these two bundles is isomorphic to $H(V_\mathbb{R})\oplus H(V_\mathbb{R})$ ?

Answer 1

As a real vector bundle the pullback has fiber over $x \in P(V_\mathbb{R})$ equal to $F_x \oplus i F_x$ where $F_x$ is the tautological fiber over $x$. Clearly, $iF_x$ is isomorphic to $F_x$ as real vector bundles, so we have the result.