On a Conjecture of Kelly on $(1,3)$-representation of Sylvester Gallai Designs

We give an exact criterion of a conjecture of L.M.Kelly to hold true which is stated as follows. If there is a finite family $\mathrm{\Sigma }$ of mutually skew lines in ${\mathbb{R}}^l,l\ge 4$ such that the three dimensional affine span (hull) of every two lines in $\mathrm{\Sigma }$, contains at least one more line of $\mathrm{\Sigma }$, then we have that $\mathrm{\Sigma }$ is entirely contained in a three dimensional space if and only if the arrangement of affine hulls is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, that is, for $(2,5)$-representations of sylvester-gallai designs in ${\mathbb{R}}^6$, which is answered in the last section.