Planes

For a rank-4 matroid, a (nontrivial) plane is a rank-3 flat containing at least 4 points, i.e. meeting at least 4 parallel classes. Any three points not on a line span a rank-3 flat, so flats with only three points carry no information and are not listed. In a realization by vectors in $k^4$, the points on a plane are coplanar in projective 3-space.

The nontrivial planes are the hyperplanes with at least 4 points. Together with the lines, the loops and the parallel classes, they determine the matroid; the affine diagram shades them.