An LR pair that can be extended to an LR triple

Fix an integer $d\ge 0$, 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 $V = \sum_{i=0}^d V_i$(directsum)\mathrm{.}Consider$$\mathbb{F}$$-lineartransformations$A$,$B$from$V$to$V$\mathrm{.}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 $A,B,C$.