My question about $E_6$ survived, so I post next episode. From the Yokota book I found out that there is $-1$ in $E_7$ Lie group. This book defines Lie group $E_7$ using 56-dimensional Freudenthal vector space over complex numbers. I prefer to use 28-dimensional quaternion space for representation of $E_7$. See also Wilson papers with other authors: symplectic and quaternionic.

The 32-dimensional symmetric space $E_{VI}$ is embedded in Lie group $E_7$ as following set $\{x^2=1, \text{x fix 16-dimensional space}\}$. There is also another embedding

$-E_{VI}$=$\{x^2=1, \text{x fix 12-dimensional space}\}$.

Spaces 70-dimensional $E_V$ and 54-dimensional $E_{VII}$ are represented as conjugacy classes in the set $\{x^2=-1\}$. $E_{VII}$=$\frac{E_7}{E_6\times S^1}$ is complex manifold and it correspond to involution $\mathbb {\tilde H}\otimes \mathbb O$. Tilde sign reverse second component in Cayley-Dickson definition of multiplication. Involution $\mathbb H\otimes \mathbb {\tilde O}$ correspond to $E_{VI}$. Product of these involutions gives $\mathbb {\tilde H}\otimes \mathbb {\tilde O}$ and $E_V$ which is split case.

One dimensional versions (over $\mathbb H\otimes \mathbb O$) of above symmetric spaces are $G_{2,10}^+\times S^2$, $G_{4,8}^+$, $G_{6,6}^+\times S^2$. The dimensions are 22, 32, 38 and are smaller by 32 then first ones.

One will ask what is my question. I ask for confirmation that above reasoning might be correct. I am planning to define exceptional Lie groups using Clifford algebras, where all compact and non-compact Lie algebras of "dimension one" can be defined using $\mathbb A \otimes \mathbb B$. What is remaining is to glue three copies using triality. This is already done in Barton-Sudbery paper. What is still remaining is to continue for Lie groups and understand the geometry.

EDIT October 2016 I paste following picture illustrating symmetric spaces for $E_7$. I leave pictures without explanation - Barton-Sudbery is the explanation.

First picture show $E_7$ compact and $E_{VI}$ first presentation. E_7 compact

Second picture show $E_{VII}$, $E_V$, $E_{VI}$ second presentation. enter image description here


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