For three copies of $\mathbb{P}^1$, say $\mathbb{P}(A)$, $\mathbb{P}(B)$ and $\mathbb{P}(C)$, with respective universal invertible quotients, $$q_A:A\otimes_k\mathcal{O}_{\mathbb{P}(A)}\to \mathcal{O}_{\mathbb{P}(A)}(1), $$ $$q_B:B\otimes_k\mathcal{O}_{\mathbb{P}(B)}\to \mathcal{O}_{\mathbb{P}(B)}(1), $$ $$q_C:C\otimes_k\mathcal{O}_{\mathbb{P}(C)}\to \mathcal{O}_{\mathbb{P}(C)}(1), $$ on $X = \mathbb{P}(A)\times \mathbb{P}(B)\times \mathbb{P}(C)$, form the rank 3, locally free sheaf $$\mathcal{E} = \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}(1) \oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}(1)\oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}(1), $$ with the associated quotient, $$ q: (A\oplus B \oplus C)\otimes_k \mathcal{O}_X \to \mathcal{E}.$$ This quotient defines an induced morphism, $$ f :\mathbb{P}_X(\mathcal{E}) \to \mathbb{P}(A\oplus B\oplus C).$$

Edit. It turns out that what follows is simply the realization of a blowing up of $\mathbb{P}^5$ as a $\mathbb{P}^2$-bundle over $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$. This is the blowing up that the OP already knows about, not the blowing up that the OP is asking about.

For three copies of $\mathbb{P}^1$, say $\mathbb{P}(A)$, $\mathbb{P}(B)$ and $\mathbb{P}(C)$, with respective universal invertible quotients, $$q_A:A\otimes_k\mathcal{O}_{\mathbb{P}(A)}\to \mathcal{O}_{\mathbb{P}(A)}(1), $$ $$q_B:B\otimes_k\mathcal{O}_{\mathbb{P}(B)}\to \mathcal{O}_{\mathbb{P}(B)}(1), $$ $$q_C:C\otimes_k\mathcal{O}_{\mathbb{P}(C)}\to \mathcal{O}_{\mathbb{P}(C)}(1), $$ on $X = \mathbb{P}(A)\times \mathbb{P}(B)\times \mathbb{P}(C)$, form the rank 3, locally free sheaf $$\mathcal{E} = \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(A)}(1) \oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(B)}(1)\oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(C)}(1), $$ with the associated quotient, $$ q: (A\oplus B \oplus C)\otimes_k \mathcal{O}_X \to \mathcal{E}.$$ This quotient defines an induced morphism, $$ f :\mathbb{P}_X(\mathcal{E}) \to \mathbb{P}(A\oplus B\oplus C).$$ The three obvious quotients, $$ r_A : \mathcal{E} \to \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(A)}(1), \ \ r_B : \mathcal{E} \to \text{pr}_{\mathbb{P}(B)}^*\mathcal{O}_{\mathbb{P}(B)}(1), \ \ r_C : \mathcal{E} \to \text{pr}_{\mathbb{P}(C)}^*\mathcal{O}_{\mathbb{P}(C)}(1), $$ which in turn defines three sections, $$ s_A: X\to \mathbb{P}_X(\mathcal{E}), \ \ s_B : X\to \mathbb{P}_X(\mathcal{E}),\ \ s_C:X\to \mathbb{P}_X(\mathcal{E}), $$ of the projection to $X$. The pairwise spans of these three sections are sub-$\mathbb{P}^1$-bundles, $$ \mathbb{P}_X(\mathcal{E}_{B,C}), \mathbb{P}_X(\mathcal{E}_{A,C}), \mathbb{P}_X(\mathcal{E}_{A,B}). $$ The images of these three subbundles are contracted under $f$ to the three $\mathbb{P}^3$s, $$\mathbb{P}(B\oplus C), \ \ \mathbb{P}(A\oplus C), \ \ \mathbb{P}(A\oplus B).$$ Thus $f$ is a birational, projective morphism that is an isomorphism over the complement of $$\mathbb{P}(B\oplus C) \cup \mathbb{P}(A\oplus C) \cup \mathbb{P}(A\oplus B).$$