Leonard triples of $q$-Racah type

Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. Pick a nonzero $q\in \mathbb{F}$ such that $q^4≠1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W,W^{\mathrm{\prime }},W^{\mathrm{\prime }\mathrm{\prime }}$ in ${\rm End}(V)$suchthat(i)$A$commuteswith$W$and$W^{-1}BW-C$;(ii)$B$commuteswith$W’$and$(W’)^{-1}CW’-A$;(iii)$C$commuteswith$W’‘$and$(W’‘)^{-1}AW’‘-B$\mathrm{.}Moreovereachof$W,W’, W’‘$ is unique up to multiplication by a nonzero scalar in $\mathbb{F}$. We show that the three elements $W’W, W’‘W’, WW’'$ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained.