# Freudenthal geometries for exceptional simple Lie groups

When reading answer to this question I recall Freudenthal, Lie groups and foundations of geometry, 1964. In chapter 4 he describes 2-dim elliptic geometry, 2-dim projective geometry, 5-dim symplectic geometry and meta-symplectic geometry for the rows of his magic square. The details are explained in his Beziehungen, I-XI, 1954-1963. All of them can be found in selected papers forming 130 pages in total. He describes points, lines, planes and symplecta for respective geometries and relations between them.

The mentioned answer on mathoverflow refer to Omoda, 2000. I can also add Yokota monograph, 2009 which defines all exceptional Lie groups using Freudenthal method.

My question is following. Has anyone understood geometries described by Freudenthal ? If yes, then where I can read about it in English ? I tried to read two parts of Freudenthal's Beziehungen few years ago but I was far from understanding it. Maybe I will try again. The new papers which appeared after year 2000 does not explain much for me.

To give example following formulas for reflections in points of octonionic projective plane $\mathbb OP^2$ can be found in Atsuyama, 1977

$\Phi(A)X=16A\times(A\times X)+4A\circ X-3X = X-4A\circ X+4(A,X)A$

It is originally due to Freudenthal, I believe. For definition of cross product, multiplication in Jordan algebra and scalar product please look into this paper. Map $\Phi$ defines embedding of projective plane $\mathbb OP^2$ into $F_4$.

I have used the first formula in GAP scripts to investigate finite subgroups of $F_4$. Using basis in Jordan algebra $h_3\mathbb O$ we can convert such reflection to $27 \times 27$ matrix. These scripts work and I was able to obtain constellation of 819 points corresponding to finite subgroup $^3D_4(2)$ of $F_4$ Lie group.

EDIT

Yesterday I have read McCrimmon, Jordan Algebras, 1978. I enjoyed it. I found out that octonionic projective plane was discovered by Moufang in 1933. It was independently coordinatized by Jordan in 1949 and Freudenthal in 1951 by rank one elements in exceptional Jordan algebra $J=h_3\mathbb O$. Next Freudenthal developed algebraic and geometric methods for all other exceptional Lie groups.

After reading Baez, Octonions and parts of Rosenfeld books, I think that more appropriate way is to look at exceptional Lie groups as automorphisms of some "projective planes". The problem is how to define such projective plane. For example Baez has defined bi-octonionic projective plane as quotient of $E_6$ by proper subgroup. Lie group $E_6$ in turn is defined as adjoint group for Lie algebra $e_6$. It is not satisfactory for me. We should rather define first bi-octonionic projective plane using product of composition algebras $\mathbb C \otimes \mathbb O$ as Rosenfeld did.

Alternatively we could first define exceptional Lie group as matrix group. Next projective plane is defined as conjugacy class of the set $\{x^2=1\}$. See this question and this and this.

Then we can do geometry in such projective planes. How such geometry relate to the works of Freudenthal ?