  A generalized Euler parameterization of a compact Lie group is a way for
parameterizing the group starting from a maximal Lie subgroup, which allows a
simple characterization of the range of parameters. In the present paper we
consider the class of all compact connected Lie groups. We present a general
method for realizing their generalized Euler parameterization starting from any
symmetrically embedded Lie group. Our construction is based on a detailed
analysis of the geometry of these groups. As a byproduct this gives rise to an
interesting connection with certain Dyson integrals. In particular, we obtain a
geometry based proof of a Macdonald conjecture regarding the Dyson integrals
correspondent to the root systems associated to all irreducible symmetric
spaces. As an application of our general method we explicitly parameterize all
groups of the class of simple, simply connected compact Lie groups. We provide
a table giving all necessary ingredients for all such Euler parameterizations.
