  In this article we provide a detailed description of a technique to obtain a
simple parametrization for different exceptional Lie groups, such as G2, F4 and
E6, 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 G2(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 G2/SO(4), G2/SU(3), F4/Spin(9),
E6/F4 and G2(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 G2/SO(4), we have been able to compute an Einstein metric and the vielbein.
The relevance of these results in physics is discussed.
