  We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the
form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff
dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This
result was also recently proved by F\"assler and Orponen using different
techniques. Our method combines induction on scales with a special structural
property of $SL_2$ Kakeya sets, which says that locally such sets look like the
pre-image of an arrangement of plane curves above a special type of map from
$\mathbb{R}^3$ to $\mathbb{R}^2$, called a twisting projection. This reduces
the study of $SL_2$ Kakeya sets to a Kakeya-type problem for plane curves; the
latter is analyzed using a variant of Wolff's circular maximal function.
