An LR pair that can be extended to an LR triple

Abstract: Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence ${V_i}{i=0}^d$ of $1$-dimensional $\mathbb{F}$-subspaces of $V$ such that $V = \sum{i=0}^d V_i$ (direct sum). Consider $\mathbb{F}$-linear transformations $A$, $B$ from $V$ to $V$. Then $A,B$ is called an LR pair whenever there exists a decomposition ${V_i}{i=0}^d$ of $V$ such that $A V_i = V{i-1}$ and $B V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. By an LR triple we mean a $3$-tuple $A,B,C$ of $\mathbb{F}$-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 $A,B,C$.