In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak{ f}_4$. Cartan's formula is written in the standard Cartesian coordinates in $\mathbb{ R}^{ 15}$. In the present paper we explain how to find analogous formula for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak{ n}({ \mathcal D})$ is constant and 2-step graded, $\mathfrak{ n}({ \mathcal D})=\mathfrak{ n}_{ -2}\oplus\mathfrak{ n}_{ -1}$. The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho,\mathfrak{ n}_{ -1})$ and $(\tau,\mathfrak{ n}_{ -2})$ of a Lie algebra $\mathfrak{ n}_{ 00}$ contained in the $0$th order Tanaka prolongation $\mathfrak{ n}_0$ of $\mathfrak{ n}({ \mathcal D})$. Numerous examples are provided, with particular emphasis on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak{ f}_4$ and $\mathfrak{ e}_6$.
Categories: math.DG math-ph math.MP