Here is another approach, not even referring to exact sequences. Let's do this torically - let use using Fulton's fan language. The fan of ${\mathbb P}^3$ is given by the cones spanned by the vectors $e_1,e_2,e_3,-e_1-e_2-e_3$ in the vector space spanned by them. The first blowup corresponds to subdividing the fan by inserting the vector $e_1+e_2$, whereas the second blowup corresponds to subdividing the fan further by inserting the vector $e_1+e_3$ - please draw a picture! Staring at the picture will tell you that this last vector will give an edge of the four cones $(e_1, e_1+e_2, e_1+e_3)$, $(e_1+e_2, e_3, e_1+e_3)$, $(e_3, -e_1-e_2-e_3, e_1+e_3)$, $(-e_1-e_2-e_3, e_1, e_1+e_3)$. The geometry of the divisor corresponding to the vector $v=e_1+e_3$, the exceptional divisor $E_2$ you are looking for, is given by projecting these four cones onto the quotient of the vector space by the span of $v$; you can arrange this simply by setting $e_1+e_3=0$ in the above cones, getting the cones $(e_1, e_1+e_2)$, $(e_1+e_2, -e_1)$, $(-e_1, -e_2)$ and $(-e_2, e_1)$ in two dimensions. This is the standard fan of ${\mathbb F}_1$.
Here is another approach, not even referring to exact sequences. Let's do this torically - let use Fulton's fan language. The fan of ${\mathbb P}^3$ is given by the cones spanned by the vectors $e_1,e_2,e_3,-e_1-e_2-e_3$ in the vector space spanned by them. The first blowup corresponds to subdividing the fan by inserting the vector $e_1+e_2$, whereas the second blowup corresponds to subdividing the fan further by inserting the vector $e_1+e_3$ - please draw a picture! Staring at the picture will tell you that this last vector will give an edge of the four cones $(e_1, e_1+e_2, e_1+e_3)$, $(e_1+e_2, e_3, e_1+e_3)$, $(e_3, -e_1-e_2-e_3, e_1+e_3)$, $(-e_1-e_2-e_3, e_1, e_1+e_3)$. The geometry of the divisor corresponding to the vector $v=e_1+e_3$, the exceptional divisor you are looking for, is given by projecting these four cones onto the quotient of the vector space by the span of $v$; you can arrange this simply by setting $e_1+e_3=0$ in the above cones, getting the cones $(e_1, e_1+e_2)$, $(e_1+e_2, -e_1)$, $(-e_1, -e_2)$ and $(-e_2, e_1)$ in two dimensions. This is the standard fan of ${\mathbb F}_1$.