How to get the integrability of a certain Hamiltonian from Temperley-Lieb algebra?

Question

In the paper https://arxiv.org/pdf/hep-th/0509069.pdf, the author mentions that a certain Hamiltonian constructed from the Temperley-Lieb algebra

$$H = -\sum_{i=1}^{N-1} e_i~~~~~~(2.2)$$

is integrable from so-called Baxterization. I did not understand how one can easily see this from the Temperley-Lieb algebra. Moreover, the author mentions in the footnote that as with any integrable model there are of course an infinite number of integrable Hamiltonians coming from the expansion of the transfer matrix.  In this paper we shall only consider the simplest one which is linear in the generators. I want to know how one can similarly construct another integrable Hamiltonian with quadratic generators.