Exceptional real Lie algebras ${\mathfrak{f}}_4$ and ${\mathfrak{e}}_6$ via contactifications

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 ${\mathfrak{e}}_6$.