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We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces of noncompact type. For SL(4, C) there are 50 inequalities none of them redundant by (KTW). For both Sp(6, C) and Spin(7, C) there are 135 inequalities of which 24 are trivially redundant in the sense that they follow from the inequalities defining the Weyl chamber . There are 9 more redundant inequalities for each of these two groups. One interesting feature is that these inequalities do not occur for the other system (and consequently must be redundant because the two polyhedral cones are the same by Theorem 1.8). The two equal polyhedral cones D3(B3) = D3(C3) have precisely 102 facets and 51 generators (edges).

The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type