\[normrules\] immediately.

Suppose that we normalized a pair of Askey-Wilson relations to satisfy implications of Lema \[normrules\], and suppose that cases 6 and 7 do not apply. Then the only affine transformations which preserve two specified zero coefficients are affine scalings. One can use affine scalings to normalize some two nonzero coefficients to convenient values. Sections \[qpararrays2\] and \[otherpararrays\] present such normalized parameter arrays that in their Askey-Wilson relations two nonzero coefficients are basically constants. (More precisely, in the $q$-cases they depend on $q$, or equivalently, on $\beta$.) The scaling normalization is explained more thoroughly in Section \[classifics\].

Normalized $q$-parameter arrays {#qpararrays1}
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Here we present the most straightforward normalizations of the general parameter arrays in [@terwgen Section 35] with the $q$-parameter. Lemma \[awnormals\] below gives the Askey-Wilson relations for the corresponding Leonard pairs. The Askey-Wilson relations turn out to be normalized according to part 1 of Lemma \[normrules\].

\[normlps\] The parameter arrays in [[@terwgen Examples 35.2–35.8]]{} can be normalized by affine transformations $(\ref{scalingtr})$ to the following forms:

-   The $q$-Racah case: $\displaystyle\theta_i=q^{-i}+s\,q^{i+1},\quad
    \theta^*_i=q^{-i}+s^*q^{i+1}$. $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}q^{1-2i}\left(1-q^i\right)\left(1-q^{i-d-1}\right)
    \left(1-r\,q^i\right)\left(r-s\,s^*q^{d+1+i}\right)\big/r,\\
    \phi_i{&\!\!\!=\!\!\!&}q^{1-2i}\left(1-q^i\right)\nonumber
    \left(1-q^{i-d-1}\right)\left(r-s^*q^i\right)\left(s\,q^{d+1}-r\,q^i\right)\big/r.\end{aligned}$$

-   The $q$-Hahn case: $\displaystyle\theta_i=q^{-i},\quad
    \theta^*_i=q^{-i}+s^*q^{i+1}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}q^{1-2i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right)\left(1-r\,q^i\right),\\
    \phi_i{&\!\!\!=\!\!\!&}-q^{1-i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right)\left(r-s^*q^i\right).\end{aligned}$$

-   The dual $q$-Hahn case: $\displaystyle
    \theta_i=q^{-i}+s\,q^{i+1},\quad\theta^*_i=q^{-i}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}q^{1-2i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right)\left(1-r\,q^i\right),\\
    \phi_i{&\!\!\!=\!\!\!&}q^{d+2-2i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right)\left(s-r\,q^{i-d-1}\right).\end{aligned}$$

-   The 