An LR pair that can be extended to an LR triple

Fix an integer d≥0, a field F, and a vector space V over F with dimension d+1. By a decomposition of V we mean a sequence {Vi}i=0d of 1-dimensional F-subspaces of V such that V=∑i=0dVi (direct sum). Consider F-linear transformations A, B from V to V. Then A,B is called an LR pair whenever there exists a decomposition {Vi}i=0d of V such that AVi=Vi−1 and BVi=Vi+1 for 0≤i≤d, where V−1=0 and Vd+1=0. By an LR triple we mean a 3-tuple A,B,C of F-linear transformations from V to V such that any two of them form an LR pair. In the present paper, we consider how an LR pair A,B can be extended to an LR triple A,B,C.