The Q = 2, 3, 4 open Potts matrices satisfy Temperley–Lieb

Let $P=Q^{-1}{\sum }_{r=0}^{Q-1}{M}_Q^r=\mathrm{\mid }+⟩⟨+\mathrm{\mid }$, where $\mathrm{\mid }+⟩=Q^{-1\mathrm{/}2}{\sum }_a\mathrm{\mid }a⟩$, and let $D={\sum }_a\mathrm{\mid }aa⟩⟨aa\mathrm{\mid }$. The finite geometric sum gives $P^2=P$; orthogonality of the $R_Q$ eigenvalues gives $Q^{-1}{\sum }_r{R}_{Q,1}^r{R}_{Q,2}^{-r}=D$, since its action on $\mathrm{\mid }ab⟩$ is $Q^{-1}{\sum }_r{\omega }_Q^{r(a-b)}\mathrm{\mid }ab⟩$, equal to $1$ if $a=b$ and $0$ otherwise. Thus odd and even generators are $\sqrt{Q}{P}_j$ and $\sqrt{Q}{D}_{j,j+1}$. Their idempotence gives $e_i^2=\sqrt{Q}{e}_i$.

On three sites, $P_1{D}_{12}{P}_1=Q^{-1}{P}_1$ and $D_{12}{P}_2{D}_{12}=Q^{-1}{D}_{12}$. For the first identity, $P\mathrm{\mid }a⟩=Q^{-1\mathrm{/}2}\mathrm{\mid }+⟩$ and $⟨a\mathrm{\mid }P\mathrm{\mid }a⟩=Q^{-1}$, so summing $P\mathrm{\mid }a⟩⟨a\mathrm{\mid }P$ over the second-site labels yields $Q^{-1}{P}_1$. For the second, $P_2\mathrm{\mid }a⟩=Q^{-1}{\sum }_b\mathrm{\mid }b⟩$ and $D_{12}$ retains only the term $b=a$, giving coefficient $Q^{-1}$ on each $\mathrm{\mid }aa⟩$. Multiplying each identity by $(\sqrt{Q}{)}^3$ proves both adjacent triple relations.

Odd generators on distinct sites commute because they act on disjoint factors. Every even generator is a sum of products of the diagonal matrices $R_Q$ on its two sites, so any two even generators commute, including when their bonds overlap in one site. An odd and an even generator whose indices differ by more than one have disjoint supports, so commute. Finally $\sqrt{2}=2\cos (\pi \mathrm{/}4)$, $\sqrt{3}=2\cos (\pi \mathrm{/}6)$, and $2=2\cos 0$, verifying the three declared $q$ normalizations. These are exactly the relations in [F2].