  In contrast to SPD matrices, few tools exist to perform Riemannian statistics
on the open elliptope of full-rank correlation matrices. The quotient-affine
metric was recently built as the quotient of the affine-invariant metric by the
congruence action of positive diagonal matrices. The space of SPD matrices had
always been thought of as a Riemannian homogeneous space. In contrast, we view
in this work SPD matrices as a Lie group and the affine-invariant metric as a
left-invariant metric. This unexpected new viewpoint allows us to generalize
the construction of the quotient-affine metric and to show that the main
Riemannian operations can be computed numerically. However, the uniqueness of
the Riemannian logarithm or the Fr{\'e}chet mean are not ensured, which is bad
for computing on the elliptope. Hence, we define three new families of
Riemannian metrics on full-rank correlation matrices which provide Hadamard
structures, including two flat. Thus the Riemannian logarithm and the
Fr{\'e}chet mean are unique. We also define a nilpotent group structure for
which the affine logarithm and the group mean are unique. We provide the main
Riemannian/group operations of these four structures in closed form.
