  We explicitly calculate the triangle inequalities for the group PSO(8).
Therefore we explicitly solve the eigenvalues of sum problem for this group
(equivalently describing the side-lengths of geodesic triangles in the
corresponding symmetric space for the Weyl chamber-valued metric). We then
apply some computer programs to verify two basic questions/conjectures. First,
we verify that the above system of inequalities is irredundant. Then, we verify
the ``saturation conjecture'' for the decomposition of tensor products of
finite-dimensional irreducible representations of Spin(8). Namely, we show that
for any triple of dominant weights a, b, c such that a+b+c is in the root
lattice, and any positive integer N, the tensor product of the irreducible
representations V(a) and V(b) contains V(c) if and only if the tensor product
of V(Na) and V(Nb) contains V(Nc).
