  Let $M^3 \subset \mathbb{C}^2$ be a $\mathcal{C}^\omega$ Levi nondegenerate
hypersurface. In the literature, Cartan-Moser chains are detected from rather
advanced considerations: either from the construction of a Cartan connection
associated with the CR equivalence problem; or from the construction of a
formal or converging Poincar\'e-Moser normal form.
  This note provides an alternative direct elementary construction, based on
the inspection of the Lie prolongations of $5$ infinitesimal holomorphic
automorphisms to the space of second order jets of CR-transversal curves.
Within the $4$-dimensional jet fiber, the orbits of these $5$ prolonged fields
happen to have a simple cubic $2$-dimensional degenerate exceptional orbit, the
chain locus: \[ \Sigma_0 \,:=\, \big\{ (x_1,y_1,x_2,y_2) \in \mathbb{R}^4
\colon\,\, x_2 = -2x_1^2y_1-2y_1^3,\,\,\, y_2 = 2x_1y_1^2 + 2x_1^3 \big\}. \]
By plain translations, we may capture all points by working only at one point,
the origin, and computations, although conceptually enlightening, become
disappointingly simple.
