Lines

For a rank-3 matroid, a (nontrivial) line is a rank-2 flat containing at least 3 points, i.e. meeting at least 3 parallel classes. Any two points span a rank-2 flat, so flats with only two points carry no information and are not listed. In a realization by vectors in $k^3$, the points on a line are collinear in the projective plane.

A rank-3 matroid is determined by its nontrivial lines together with its loops and parallel classes, which is what its affine diagram shows.