<B>Edit.</B>  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 <B>already</B> knows about, not the blowing up that the OP is <B>asking</b> 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 
$$\mathbb{P}(B\oplus C) \cup \mathbb{P}(A\oplus C) \cup \mathbb{P}(A\oplus B).$$