Line Spreads That Produce Projective Planes

We explicitly classify those line spreads of projective 5-space over a field that have the property that the given spread induces a spread in the 3-space generated by any pair of spread lines. We determine their fix groups and conclude that there exist such spreads with a trivial fix group. Also, we characterise regular line spreads among all line spreads of projective 3-space by their projectivity group and also by a weakening of the regularity condition.