Let $X$ be a smooth cubic threefold over $\mathbb{C}$ and let $L \subset X$ be a line.

The projection from $L$ yields a rational map $X \dashrightarrow \mathbb{P}^2$. Resolving the indeterminacy by blowing up $L$, we obtain a conic bundle morphism $\pi: \tilde{X} \to \mathbb{P}^2$.

Consider the relative anticanonical bundle $\omega_{\pi}^{-1}$. By standard properties of conic bundles we obtain an embedding $$\tilde{X} \hookrightarrow \mathbb{P}(\mathcal{E})$$ which respects $\pi$. Here $\mathcal{E} := \pi_*\omega_{\pi}^{-1}$ is a vector bundle of rank $3$ on $\mathbb{P}^2$ and $\mathbb{P}(\mathcal{E}) \to \mathbb{P}^2$ is the associated $\mathbb{P}^2$-bundle over $\mathbb{P}^2$.

Is there is an explicit description of the vector bundle $\mathcal{E}$? For example, is it independent of $X$ or $L$? Is it determined by the normal bundle of $L$? Is it a direct sum of lines bundles?

Note that $\mathbb{P}(\mathcal{E}) \cong \mathbb{P}(\mathcal{E} \otimes \mathcal{O}(k))$ for any $k \in \mathbb{Z}$. In particular, I am quite happy with determining $\mathcal{E}$ up to twist by a line bundle.