Generating 4-connected even triangulations on the sphere

In this paper, we prove that every 4-connected even triangulation on the sphere can be obtained from the octahedron by a sequence of two kinds of transformations called a 4-splitting and a twin-splitting, only through 4-connected even triangulations. As a corollary, taking a dual, we can also generate all cyclically 4-edge-connected cubic bipartite graphs on the sphere.