Prolongation of $(8,15)$-Distribution of Type $F_4$ by Singular Curves

Cartan gives the model of $(8, 15)$-distribution with the exceptional simple Lie algebra $F_4$ as its symmetry algebra in his paper (1893), which is published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from viewpoints of sub-Riemannian geometry and geometric control theory.Then we construct the prolongation of Cartan's model based on the data related to its singular curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part of the graded Lie algebra $F_4$.