ADAMS-IWASAWA $\mathcal{N} = 8$ BLACK HOLES
We study some of the properties of the geometry of the exceptional Lie group E7(7), which describes the U-duality of the [Formula: see text], 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 [Formula: see text], 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 [Formula: see text] as scalar configuration of the "large" ⅛-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 ℤ4, 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.