Applying Lie's elementary theory for appropriate prolongations to jet spaces of orders 1 and 2, we show that any real analytic hypersurface M⁵ in C³ which is 2-nondegenerate of constant Levi rank 1 carries two sorts of Cartan-Moser chains, that are of orders 1 and 2.

Integrating and straightening any given order 2 chain passing through any point p in M to be the v-axis in coordinates (z, s, w = u + i v) centered at p, without setting up the formal theory in advance, we show that there exists a convergent change of complex coordinates (z, s, w) --> (z', s', w') fixing the origin in which the chain is the v-axis and in which M has a certain explicit Poincaré-Moser reduced equation.

The values at the origin of Pocchiola's two primary invariants are proportional to two specific Taylor coefficients of this normal form.

Zhangchi Chen provided the Maple figures of Sections 8 and 9.