  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 $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has
$q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\rm
End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$
commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and
$(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ 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.
