A smooth quasi-homogeneous Legendrian 8-fold.

I will explain a construction of a new example of smooth Legendrian
8-fold, which turns out to have some very interesting properties. It is a Fano variety of Picard number 1. Also it is quasihomogeneous and it is a compactification of SL(3). A careful analysis of its
properties leads to generalisations that can be applied to three different subjects of algebraic geometry:
1) This was the first step towards a construction of many examples of non homogeneous Legendrian varieties.
2) We can give a connection between so called sub-adjoint varieties
related to the exceptional groups F4, E6, E7, E8.
3) Also we can give an easy description of a smooth compactification of SL(4).