cases, there is a unique affine translation to make the normalization. In the last $2$ cases, there are infinitely many normalizations by affine translations.

[[**Proof.** ]{}]{}The first 4 cases are straightforward, including the uniqueness statement. If $\beta=2$, $\gamma=0$, $\gamma^*=0$, the new Askey-Wilson relations (\[newaw\]) are $$\begin{aligned}
 \textstyle
AW\left( 2,\,0,\,0,\,\varrho\,t^2,\,\varrho^*t^*{}^2,\,
\omega\,tt^*\!,\,\big(\eta-\omega\,a-\varrho\,a^*\big)\,t^2t^*,\,
\big(\eta^*-\varrho^*a-\omega\,a^*\big)\,t\,t^*{}^2\right),\end{aligned}$$ where $a=c/t$ and $a^*=c^*/t^*$. To set the last two parameters to zero, we have to solve two linear equations in $a,a^*$. If we have $\det{\omega\;\;
\varrho\choose\,\varrho^*\; \omega}\neq 0$, the solution is unique. Otherwise we have either infinitely many or none solutions, which leads us to the last two cases. [$\Box$]{}\
As it turns out, cases 6 and 7 of Lemma \[normrules\] do not occur for Askey-Wilson relations satisfied by Leonard pairs if $d\ge 3$. See part 3 of Theorem \[classth\] below.

In Section \[qpararrays1\], we normalize the general $q$-parameter arrays in Terwilliger’s classification [@terwgen Section 35] with most handy changes in the explicit expressions. We use the following simplest action of (\[afftr\]) on parameter arrays, consistent with the transformation of Leonard pairs: $$\label{scalingtr}
\theta_i\mapsto t\,\theta_i+c,\qquad \theta^*_i\mapsto t^*\theta^*_i+c^*,
\qquad \varphi_i\mapsto t\,t^*\varphi_i, \qquad \phi_i\mapsto t\,t^*\phi_i.$$ It turns out that the corresponding Askey-Wilson relations follow the specification of part 1 of Lemma 