# Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$

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.