  It is known that hermitean random matrix ensembles can be identified with
symmetric coset spaces of Lie groups, or else with tangent spaces of the same.
This results in a classification of random matrix ensembles as well as
applications in practical calculations of physical observables. In this paper
we show that a large number of non-hermitean random matrix ensembles defined by
physically motivated symmetries - chiral symmetry, time reversal invariance,
space rotation invariance, particle-hole symmetry, or different reality
conditions - can likewise be identified with symmetric spaces. We give explicit
representations of the random matrix ensembles identified with lateral algebra
subspaces, and of the corresponding symmetric subalgebras spanning the group of
invariance. Among the ensembles listed we identify as special cases all the
hermitean ensembles identified with Cartan classes of symmetric spaces and the
three Ginibre ensembles with complex eigenvalues.
