Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience.

$F_{I}^{28}\subset E_{II}^{40}\subset E_{VI}^{64} \subset E_{IX}^{112}$

It corresponds to following chain of algebras:

$\mathbb {\tilde O} \subset \mathbb C \otimes \mathbb {\tilde O}\subset \mathbb H \otimes \mathbb {\tilde O}\subset \mathbb O \otimes \mathbb {\tilde O}$

$\mathbb {\tilde O}$ is algebra of split octonions.

One can see that codimensions of above three inclusions are $12$, $24$, $48$. The above symmetric spaces can be embedded in corresponding Lie groups represented as matrices

$M_{27}\mathbb R\subset M_{27}\mathbb C\subset M_{28}\mathbb H\subset M_{248}\mathbb R$

The above symmetric spaces then are conjugacy classes of involutions which fix 12-dimensional space: real, complex, quaternionic and 12*8=96 real dimensions.

Intersection of $F_{I}$ with $Spin_9$, $Spin_8$, $Spin_7$, $G_2$ is respectively $G_{4,5}^+$, $G_{4,4}^+$, $G_{4,3}^+$ and 8-dimensional $G=G_2/SO_4$.

Intersections of first chain with $Spin_9$, $Spin_{10}$, $Spin_{12}$, $Spin_{16}$ are: $G_{4,5}^+\subset G_{4,6}^+\subset G_{4,8}^+\subset G_{4,12}^+$ giving codimensions: $4$, $8$, $16$.

Question: What explanation to the dimensions of above spaces can be given ? I wonder also how we could obtain the quaternionic structure from $SU_2=Sp_1$ factor of $H$ in $G/H$ in each case?


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