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In Cartan's PhD thesis, there is a formula defining a certain rank 8 vectordistribution in dimension 15, whose algebra of authomorphism is the split realform of the simple exceptional complex Lie algebra $\mathfrak{f}_4$. Cartan'sformula is written in the standard Cartesian coordinates in $\mathbb{R}^{15}$.In the present paper we explain how to find analogous formula for the flatmodels 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 linearalgebraic 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 distributionswith symmetries being real forms of simple exceptional Lie algebras$\mathfrak{f}_4$ and $\mathfrak{e}_6$.

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