  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 $\mathfrak{e}_6$.
