  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 $\Sigma$ of mutually skew lines
in $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) of
every two lines in $\Sigma$, contains at least one more line of $\Sigma$, then
we have that $\Sigma$ 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.
