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In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $E_6$ Lie group. Spaces $E_{II}$ and $E_{III}$ can be embedded as set $M=\{x^2=1\}$ having $12$ and $16$ dimensions of eigenspaces with eigenvalue $-1$. Consider set $P=\{x^2 \in E_{III}\}$ in Lie group $E_6$. Is there chance to find $E_I$ or $E_{IV}$ in $P$ ?

The hint is $Spin_{10}$ subgroup and grassmanians which can be found in set $\{x^2=-1\}$ in Clifford algebra.

This is duplicate question of the one on math.stackexchange.

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