Proposition. Let $l_1,l_2,l_3$ be lines in $P(β^4)$ that do not intersect pairwise. Then there are an infinite number of lines in $l$ in $P(β^4)$ that intersect $l_1,l_2,l_3$.
Proof. Let $l_j=P(U_j),j=1,2,3$. Then $U_1,U_2,U_3$ are vector subspaces of $β^4$, $\dim U_j=2$.
Since $l_1\cap l_2=\emptyset$, $U_1\cap U_2=\{0\}$. Hence $β^4=U_1\oplus U_2$, counting dimensions.
Let $v\in U_3$ be non-zero, then $v=u_1+u_2,u_1\in U_1,u_2\in U_2$.
If $u_1=0$ then $[v]\in l_2\cap l_3$β
If $u_2=0$ then $[v]\in l_1\cap l_3$β
Let $U_4=\langle u_1,u_2\rangle\leβ^4$. Then $\dim U_4=2$. So $l_4=P(U_4)$ is a line in $P(β^4)$.
Also, $l_4$ contains $[u_1]\in l_1$ and $[u_2]\in l_2$ and $[u_1+u_2]=[v]\in l_3$.
So $l_4$ intersects $l_1,l_2,l_3$. Unique such line intersecting $l_3$ in the point $[v]$. By construction, there exists such a line through each point $[v]$ in $l_3$, so there are infinitely many such lines $l$. β‘
Here is a more difficult proof.
Desarguesβ theorem
Let $P,Q_1,Q_2,Q_3,R_1,R_2,R_3$ be distinct points in a projective plane $P(V)$, such that the lines $Q_1R_1,Q_2R_2,Q_3R_3$ are distinct and concurrent at $P$.
Let $S_1$ be the intersection of $Q_2Q_3,R_2R_3$
Let $S_2$ be the intersection of $Q_3Q_1,R_3R_1$
Let $S_3$ be the intersection of $Q_1Q_2,R_1R_2$
Then $S_1,S_2,S_3$ are collinear.
Proof.
Choose representatives $p,q_i,r_i$ such that $p=q_i+r_i$.
As $p=q_2+r_2=q_3+r_3$, put $s_1:=q_2-q_3=r_3-r_2$.
Since $Q_2,Q_3$ are distinct, $s_1$ is nonzero.
Hence $s_1$ represents a point on the line $Q_2Q_3$.
Similarly, $s_1$ represents a point on the line $R_2R_3$.
Hence $s_1$ represents $S_1$, the intersection of $Q_2Q_3,R_2R_3$.
Similarly, put $s_2:=q_3-q_1=r_1-r_3,s_3:=q_1-q_2=r_2-r_1$, and then $s_2,s_3$ represent $S_2,S_3$,
Now, $s_1+s_2+s_3=(q_2-q_3)+(q_3-q_1)+(q_1-q_2)=0$.
Hence $S_1,S_2,S_3$ are collinear, as $S_1,S_2,S_3$ are linearly dependent. β‘