Such a quotient space is a symmetric space, and so one can generically take Φ Φ = G / H where H ⊂ ⊂ G is the maximal subgroup of G that is invariant under the Cartan involution. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on G to a metric on G / H or as a pullback of the Maurer–Cartan form.