The four triples or triangles (a8, a1, a2), (a2, a3, a4), (a4, a5, a6), (a6, a7, a8) form a 4-cycle-triple block, and the four triangles (a6, a1, a2), (a2, a3, a4), (a4, a5, a6), (a6, a7, a8) form a kite-triple block. The "interior" of the 4- cycle-triple block consists of the 4-cycle (a2, a4, a6, a8), and the "kite interior" of the kite-triple block consists of the kite (a2, a4, a6) - a8 (consisting of a triangle with pendant edge). A 3λ-fold decomposition of Kn into 4- cycle-triple blocks (or into kite-triple blocks) such that the "interiors" of each block form a λ-fold 4-cycle system (or kite system) is said to be a perfect 3λ-fold decomposition of Kn. We find such perfect 4-cycle-triple and kite-triple systems for all λ and n.