In this article we provide a detailed description of a technique to obtain a simple parameterization for different exceptional Lie groups, such as G{sub 2}, F{sub 4} and E{sub 6}, based on their fibration structure. For the compact case, we construct a realization which is a generalization of the Euler angles for SU(2), while for the non compact version of G{sub 2(2)}/SO(4) we compute the Iwasawa decomposition. This allows us to obtain not only an explicit expression for the Haar measure on the group manifold, but also for the cosets G{sub 2}/SO(4), G{sub 2}/SU(3), F{sub 4}/Spin(9), E{sub 6}/F{sub 4} and G{sub 2(2)}/SO(4) that we used to find the concrete realization of the general element of the group. Moreover, as a by-product, in the simplest case of G{sub 2}/SO(4), we have been able to compute an Einstein metric and the vielbein. The relevance of these results in physics is discussed.