I am interested in the Lie algebra decompositions of the isometry group of the real Grassmanianns (seen as symmetric spaces). More precisely, let $Gr(r,n)=O(n)/(O(r)\times O(n-r))$. Can someone indicate some references where they compute the corresponding Iwasawa decomposition of $\mathfrak{so}(n)$? I am especially interested in the root and the restricted roots.
Any help is appreciated.