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EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS
$\mathfrak {f}_4$ AND 
$\mathfrak {e}_6$ VIA CONTACTIFICATIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 1 / January 2025
- Published online by Cambridge University Press:
- 03 July 2024, pp. 157-201
- Print publication:
- January 2025
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- Article
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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 formulae 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 put on the distributions with symmetries being real forms of simple exceptional Lie algebras
$\mathfrak {f}_4$ and 
$\mathfrak {e}_6$.
