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

Abstract

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 $\mathfrak{e}_6$.