  We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of
the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show
that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup
\mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and
Zahl.
  The $SL(2)$ lines can be identified with horizontal lines in the first
Heisenberg group, and we obtain the main result as a corollary of a more
general statement concerning unions of horizontal lines. This statement is
established via a point-line duality principle between horizontal and conical
lines in $\mathbb{R}^{3}$, combined with recent work on restricted families of
projections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang.
  Our result also has a corollary for Nikodym sets associated with horizontal
lines, which answers a special case of a question of Kim.
