The dga of planar loops when $2n=4$

The dga of planar loops was introduced in recent work of the author, Boyd, Randal-Williams, and Sroka, where a minimal model for it was given. This dga enjoys two natural `reflection’ involutions. In the first nontrivial case, $2n=4$, we give a new model which incorporates these involutions, as well as a more explicit description of the existing model.