The derivation of smooth cosmic billiard solutions by means of the compensator method, introduced by us sometimes ago, is extended to the case of supergravity with non-maximal supersymmetry. Here a new key feature is provided by the non-maximal split nature of the scalar coset manifold. To deal with this, one has to consider the theory of Tits-Satake projections leading to maximal split projected algebras, where the compensator method can be successfully applied and interesting solutions that display several smooth bounces can be derived. The generic bouncing feature of all exact solutions can thus be checked. From the analysis of the Tits-Satake projection emerges a regular scheme applicable to all non-maximal supergravity models and in particular a challenging so far unobserved structure, that of the paint group G paint. This latter, which is preserved through dimensional reduction, provides a powerful tool to codify solutions of the exact supergravity theories in terms of solutions of their Tits-Satake projected partners, which are much simpler and manageable. It appears that the dynamical walls on which the cosmic ball bounces come actually in painted copies rotated into each other by the paint group. So the effective cosmic dynamics is that dictated by the maximal split Tits-Satake manifold plus paint. In the present paper we work out in all minor details the example provided by N = 6, D = 4 supergravity, whose scalar manifold is the special Kählerian SO*/SU(6) × U(1) c-mapping in D = 3 to the quaternionic E7(-5)/SO(12) × SO(3). This choice was not random. It is the next one after maximal supergravity and at the same time can be reinterpreted in the context of N = 2 supergravity. We plan indeed, in a future publication, to apply the results we obtained here, to the discussion of the Tits-Satake projection within the context of generic special Kähler manifolds. We also comment on the merging of the Tits-Satake projection with the affine Kač-Moody extension originating in dimensional reduction to D = 2 and relying on a general field-theoretical mechanism illustrated by us in a separate paper.