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We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines ofthe form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We showthat 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 andZahl.  The $SL(2)$ lines can be identified with horizontal lines in the firstHeisenberg group, and we obtain the main result as a corollary of a moregeneral statement concerning unions of horizontal lines. This statement isestablished via a point-line duality principle between horizontal and conicallines in $\mathbb{R}^{3}$, combined with recent work on restricted families ofprojections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang.  Our result also has a corollary for Nikodym sets associated with horizontallines, which answers a special case of a question of Kim.

A note on Kakeya sets of horizontal and $SL(2)$ lines