Note on the Correspondence Between Two Differential Equations with G2-symmetry

The aim of this note is to review carefully the two famous G2-models given by E. Cartan in [4] and [6] within the contemporary framework of parabolic geometries of G2-type brought by K. Yamaguchi ([24], [25], [28]). Utilizing the above framework, we describe a direct correspondence between the two model equations with G2-symmetry based on the construction of the PD manifold of second order.