3 Edit to atone for my mistakes.

The following is mostly bogus, based on my overly quick reading and misunderstanding of the question:

No. $E_7 / (SU(8)/\mu_2)$ (where $E_7$ here denotes the compact real Lie group) is not a Hermitian symmetric space. You can tell, because there is no torus (copy of $U(1)$) in the center of $SU(8)$ (maybe Helgasson or Wolf is a good reference here).

The following comment does not apply, since the manifold has dimension $4k+2$ and not $4k$

Hence, by Gauduchon, Moroianu, Semmelmann (Inventiones Math., 2010), as you seem to know, this exceptional inner symmetric space carries no almost complex structure.

The following is correct:

I have no idea how to answer this question.

2 Note: I am making a trivial change in order to be able to remove an improper upvote.

No. $E_7 / (SU(8)/\mu_2)$ (where $E_7$ here denotes the compact real Lie group) is not a Hermitian symmetric space. You can tell, because there is no torus (copy of $U(1)$) in the center of $SU(8)$ (maybe Helgasson or Wolf is a good reference here).

Hence, by Gauduchon, Moroianu, Semmelmann (Invent. Inventiones Math., 2010), as you seem to know, this exceptional inner symmetric space carries no almost complex structure.

1

No. $E_7 / (SU(8)/\mu_2)$ (where $E_7$ here denotes the compact real Lie group) is not a Hermitian symmetric space. You can tell, because there is no torus (copy of $U(1)$) in the center of $SU(8)$ (maybe Helgasson or Wolf is a good reference here).

Hence, by Gauduchon, Moroianu, Semmelmann (Invent. Math., 2010), as you seem to know, this exceptional inner symmetric space carries no almost complex structure.