  We construct two parametrizations of the non compact exceptional Lie group
G=E7(-25), based on a fibration which has the maximal compact subgroup K=(E6 x
U(1))/Z_3 as a fiber. It is well known that G plays an important role in the
N=2 d=4 magic exceptional supergravity, where it describes the U-duality of the
theory and where the symmetric space M=G/K gives the vector multiplets' scalar
manifold. First, by making use of the exponential map, we compute a realization
of G/K, that is based on the E6 invariant d-tensor, and hence exhibits the
maximal possible manifest [(E6 x U(1))/Z_3]-covariance. This provides a basis
for the corresponding supergravity theory, which is the analogue of the
Calabi-Vesentini coordinates. Then we study the Iwasawa decomposition. Its main
feature is that it is SO(8)-covariant and therefore it highlights the role of
triality. Along the way we analyze the relevant chain of maximal embeddings
which leads to SO(8). It is worth noticing that being based on the properties
of a "mixed" Freudenthal-Tits magic square, the whole procedure can be
generalized to a broader class of groups of type E7.
