that $\beta$ stays invariant. The affine transformations $$\label{afftr}
(A,A^*)\mapsto (tA+c,\, t^*\!A^*\!+c^*),\qquad \mbox{with}\
c,c^*,t,t^*\!\in{{\Bbb K}},\ t,t^*\neq 0,$$ can be used to normalize Leonard pairs so that their Askey-Wilson relations would have a simple form. We refer to a transformations of the form $(A,A^*)\mapsto (A+c,A^*+c^*)$ as an [*affine translation*]{}, and to a transformation of the form $(A,A^*)\mapsto (tA,\,t^*A^*)$ as an [*affine scaling*]{}. Generally, we can use an affine translation to set some two Askey-Wilson coefficients to zero, and then use an affine scaling to normalize some two nonzero coefficients. Specifically, by affine translations we can achieve the following.

\[normrules\] The Askey-Wilson relations $AW(\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*)$ can be normalized as follows:

1.  If $\beta\neq 2$, we can set $\gamma=0$, $\gamma^*=0$.

2.  If $\beta=2$, $\gamma\neq 0$, $\gamma^*\!\neq 0$, we can set $\varrho=0$, $\varrho^*=0$.

3.  If $\beta=2$, $\gamma=0$, $\gamma^*\!\neq 0$, we can set $\varrho^*=0$, $\omega=0$.

4.  If $\beta=2$, $\gamma^*\!=0$, $\gamma\neq 0$, we can set $\varrho=0$, $\omega=0$.

5.  If $\beta=2$, $\gamma=0$, $\gamma^*=0$, $\omega^2\neq\varrho\varrho^*$, we can set $\eta=0$, $\eta^*=0$.

6.  If $\beta=2$, $\gamma=0$, $\gamma^*=0$, $\mbox{\rm
    rk}{\omega\;\; \varrho\;\;\eta\choose\,\varrho^*\,\;\omega\;\eta^*}\le 1$, we can set $\eta=0$, $\eta^*=0$.

7.  Otherwise, we have $$\beta=2,\quad \gamma=0,\quad \gamma^*=0,\quad
    \omega^2=\varrho\varrho^*,\quad \mbox{\rm rk}{\omega\quad
    \varrho\quad\eta\,\choose\varrho^*\quad\!\!\omega\quad\eta^*}=2.$$ Then we can set either $\eta=0$ or $\eta^*=0$, but not both.

In the first $5$ 