The notation, ideas and to some extent the results and examples of previous papers of the author are reproved in such a way that those results become special cases of the work under consideration. The structure of the double coset decomposition \(H\setminus G/L\), when \(G\) is a reductive Lie group, is discussed in this paper. The author has succeeded in finding representatives for all the \(H-L\) double cosets of \(G\). The case, when the semisimple part \(Gs\) of \(G\) is compact, is discussed in section 3 and section 4 deals with the case of arbitrary reductive Lie groups \(G\). The author illustrates his conclusions with suitable examples from compact and noncompact Lie groups.
Finally, the author introduces the idea of equivalence of pairs of involutions on a Lie algebra \({\mathfrak g}\) as follows: If \(\sigma \), \(\sigma'\) and \(\tau\), \(\tau'\) are involutions on \({\mathfrak g}\), then the pairs \((\sigma, \tau)\) and \((\sigma',\tau')\) are equivalent if and only if there exist an automorphism \(\rho\) and an inner automorphism \(\rho_0\) of \({\mathfrak g}\) such that \(\sigma'=\rho\sigma\rho^{-1}\) and \(\tau'= \rho_0 \rho \tau \rho^{-1} \rho_0^{-1}\). By this equivalence relation the author has classified pairs of involutions for compact Lie groups and the resulting results appear in a subsequent paper announced in the reference to the present work.