The Weyl Integration Model for KAK decomposition of Reductive Lie Group
  The Weyl integration model presented by An and Wang can be effectively used to reduce the integration over $G$-space. In this paper, we construct an especial Weyl integration model for KAK decomposition of Reductive Lie Group and obtain an integration formula which implies that the integration of $L^1$-integrable function over reductive Lie group $G$ can be carried out by first integrating over each conjugacy class and then integrating over the set of conjugacy classes. 
Zhi-guang Hu and Kui-hua Yan