  We analyze the polynomial part of the Iwasawa realization of the coset
representative of non compact symmetric Riemannian spaces. We start by studying
the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and
how it determines the structure of the nilpotent subalgebras. This allows us to
compute the maximal degree of the polynomials for all faithful representations
of Lie algebras. In particular the metric coefficients are related to the
scalar kinetic terms while the representation of electric and magnetic charges
is related to the coupling of scalars to vector field strengths as they appear
in the Lagrangian. We consider symmetric scalar manifolds in N-extended
supergravity in various space-time dimensions, elucidating various relations
with the underlying Jordan algebras and normed Hurwitz algebras. For magic
supergravity theories, our results are consistent with the Tits-Satake
projection of symmetric spaces and the nilpotency degree turns out to depend
only on the space-time dimension of the theory. These results should be helpful
within a deeper investigation of the corresponding supergravity theory, e.g. in
studying ultraviolet properties of maximal supergravity in various dimensions.
