Local equivalence of some maximally symmetric $(2,3,5)$-distributions

Using a complex parametrisation of $su(2)$, we show a change of coordinates that maps the maximally symmetric rolling $(2,3,5)$-distribution to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric rolling model and the flat Cartan or Hilbert-Cartan distribution. For the maximally symmetric rolling distribution, we write down the vector fields that bracket-generate to give the split real form of the Lie algebra of $\frak{g}_2$, with two of the vector fields in the bracket-generating set given by the span of the rolling distribution.