 Solution

The two projective lines in the $i$th pair form the fibre of a morphism $f^{-1}([s_i:t_i])$ of the extended conic bundle from a line on a smooth cubic surface constructed above. Distinct points of the projective line have disjoint inverse images, soThe extension of $f$ across $l$ matters: arguing only that distinct planes through $l$ meet along $l$ would not by itself exclude intersections there. The formula $f{|}_l=[-B:A]$ excludes them as well. Lines within a single pair do intersect; the assertion concerns different pairs.