= 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, so
$$
(l_i\cup l_i')\cap(l_j\cup l_j')=\varnothing\qquad(i\ne j).
$$
The 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.
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