# Exceptional symmetric spaces embedded in exceptional Lie group

In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective spaces $AP^2$, where $A=\mathbb K\otimes \mathbb O$, where $\mathbb K= \mathbb R, \mathbb C, \mathbb H, \mathbb O$ into $F_4$, $E_6$, $E_7$, $E_8$ Lie groups.

When defining $F_4$ as automorphisms of Jordan algebra $h_3\mathbb O$ then obtained matrices are real matrices of dimension $27$. The octonion projective space $\mathbb OP^2$ can be defined as order 2 elements having dimension of $-1$ eigenspace $16$.

Similarly next spaces for $E_6$ and $E_7$ are obtained as set of matrices of order 2 having dimension 16 of eigenspace in complex and quaternion matrices of dimension $27$ and $28$ respectively. For $E_8$ we define set of order 2 elements having dimension $16*8$ of eigenspace in matrices of dimensions $31*8$.

I was able to define $AP^1$ spaces as grassmanians in Clifford Algebras $C_8$, $C_9$, $C_{10}$ and $C_{14}$ which are $M_{16}\mathbb R$, $M_{16}\mathbb C$, $M_{16}\mathbb H$ and $M_{128}\mathbb R$. Grassmanians are $G_{k,8}^+$ for k=1, 2, 4, 8.

Now there are following ideas.

1. Use above definitions in order to define exceptional Lie groups. Lie group is generated by points of $AP^2$. Usually using Jordan algebra one can define $F_4$ and $E_6$. It is hard to find good definition of $E_7$ and $E_8$ (does exist such ?).
2. Look for other exceptional symmetric spaces in the Lie group. I claim that $E_{II}$ of dimension $40$ is obtained as order 2 elements in $E_6$ reversing 12-dimensional complex subspace. Those elements can be obtained as products of "perpendicular" elements in $E_{III}$=$\mathbb C\otimes \mathbb OP^2$ (in case when their product is not in $E_{III}$).
3. Look for symmetric constellations on $AP^2$. For example I am looking for 1755 elements in $E_{III}$ generating Tits subgroup in $E_6$. See Robert Wilson.

See for example Eschenburg for similar ideas.

My question is following. Is this plan reasonable ? Is this possible to achieve and is this true $:)$ what I am saying. By $plan$ I mean defining in some algebraic way the $16-$, $32-$, $64-$ and $128-dimensional$ space in proper matrix space.

Finally I will add humorous note about multiples of 8. In year 2008 I constructed all exceptional Lie algebras in GAP using Barton-Sudbery. In year 2016 I add this post to mathoverflow. I expect in 2024 I should have some theory ready to publish :)

• This reminds my of Boris Rosenfeld, Geometry of Lie Groups, where he constructs the exceptional Lie groups as symmetries of various "projective planes", but using a generalization of the notion of projective plane which does not give projective planes in the sense of Hilbert's definition. I suppose you must have read Rosenfeld's book, but in case you haven't, you should have a look. – Ben McKay May 24 '16 at 8:37
• I don't want to vote to close, but I will say it's "unclear what you're asking". – Allen Knutson May 24 '16 at 11:12
• I clarified what my question is by editing it. Philosophically, having such embeddings we will know what are the "octonion reflections". Namely, these are points of embedded projective spaces (although someone may doubt whether these are projective spaces). – user21230 May 24 '16 at 12:37