  We study some of the properties of the geometry of the exceptional Lie group
E7(7), which describes the U-duality of the N=8, d=4 supergravity. In
particular, based on a symplectic construction of the Lie algebra e7(7) due to
Adams, we compute the Iwasawa decomposition of the symmetric space
M=E7(7)/(SU(8)/Z_2), which gives the vector multiplets' scalar manifold of the
corresponding supergravity theory.
  The explicit expression of the Lie algebra is then used to analyze the origin
of M as scalar configuration of the "large" 1/8-BPS extremal black hole
attractors. In this framework it turns out that the U(1) symmetry spanning such
attractors is broken down to a discrete subgroup Z_4, spoiling their dyonic
nature near the origin of the scalar manifold.
  This is a consequence of the fact that the maximal manifest off-shell
symmetry of the Iwasawa parametrization is determined by a completely
non-compact Cartan subalgebra of the maximal subgroup SL(8,R) of E7(7), which
breaks down the maximal possible covariance SL(8,R) to a smaller SL(7,R)
subgroup. These results are compared with the ones obtained in other known
bases, such as the Sezgin-van Nieuwenhuizen and the Cremmer-Julia /de
Wit-Nicolai frames.
