Page "Symmetric space" Paragraph 18

The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer is a Lie subalgebra of.

The second condition means that is an-invariant complement to in.

Thus any symmetric space is a reductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces.

The key feature of symmetric spaces is the third condition that brackets into.