In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines a generic algebra $\mathcal{E}_A(a_s,b_s)$. It is defined as below:
Given a commutative ring $A$ and a Coxeter group $W$ with a set of generators $S$, assume we have some parameters $a_s,b_s\in A \ (s\in S)$ subject only to the requirement that $a_s=a_t$ and $b_s=b_t$ whenever $s$ and $t$ are conjugate in $W$.
$\mathcal{E}_A(a_s,b_s)$ is a free $A$-module on the set $W$, with basis elements denoted $T_w \ (w \in W)$. It has a unique structure of associative $A$-algebra satisfying
(1)$\qquad$ $T_s T_w=T_{sw}$ $\ $ if $\ $ $l(sw)>l(w)$
(2)$\qquad$ $T_s T_w=a_s T_w + b_s T_{sw}$ $\ $ if $\ $ $l(sw)<l(w)$
where $T_1$ acts as an identity. We call it a generic algebra.
In the textbook p.150 Exercise 2 gives a presentation of the generic algebra $\mathcal{E}_A(a_s,b_s)$ as an $A$-algebra.
Exercise 2. Show that the generic algebra $\mathcal{E}_A(a_s,b_s)$ has a presentation as the $A$-algebra with $1$ generated by the elements $T_s \ (s\in S)$, subject only to the relations: ${T_s}^2=a_s T_s +b_s T_1$, $(T_s T_t)^q=(T_t T_s)^q$ $\;$if$\;$ $m(s,t)=2q < \infty $,$\;$ $(T_s T_t)^q T_s=(T_t T_s)^q T_t$ $\;$if$\;$ $m(s,t)=2q+1 < \infty $
But I don't know why these relations give a presentation of $\mathcal{E}_A(a_s,b_s)$. If we consider a free algebra $F$ over $A$ generated by elements corresponding to $W$, we will have a canonical surjection $F \twoheadrightarrow \mathcal{E}_A(a_s,b_s)$. If we deonte the kernel by $I$, I can see that $I$ includes a two-sided ideal of $F$ generated by all relations. However, I cannot prove that $I$ is included in the two-sided ideal.
Any helps will be appreciated. Thank you for reading.