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2 clarified the reductivity condition

I very much fear this answer will not satisfy you completely. Symmetric spaces comprise a very small subclass of homogeneous spaces, even if when you restrict to the hermitian ones.

One way to try to understand what is going on is to consider already the smaller class of reductive homogeneous spaces. In positive-definite signature, all homogeneous spaces are reductive, but in indefinite signature this is not necessarily the case.)

A theorem of Ambrose and Singer, reformulated by Kostant, states that a riemannian manifold is (locally) reductive homogeneous if and only if there is a metric connection such that both the torsion and curvature are parallel. The (locally) symmetric spaces are precisely those for which the torsion vanishes. This is a very strong condition.

1

I very much fear this answer will not satisfy you completely. Symmetric spaces comprise a very small subclass of homogeneous spaces, even if when you restrict to the hermitian ones.

One way to try to understand what is going on is to consider already the smaller class of reductive homogeneous spaces. A theorem of Ambrose and Singer, reformulated by Kostant, states that a riemannian manifold is (locally) reductive homogeneous if and only if there is a metric connection such that both the torsion and curvature are parallel. The (locally) symmetric spaces are precisely those for which the torsion vanishes. This is a very strong condition.