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.

- 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 ?).
- 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 $E_{III}$).
- 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 :)