Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 and p. 198]: it has generators the symbols $x_{\alpha}(a)$ ($\alpha\in\Phi$, $a\in R$) and the following defining relations, where $\alpha,\beta\in\Phi$, $a,b\in R$ and $t,u\in R^{\times}$:

(R1) $x_{\alpha}(a) x_{\alpha}(b)=x_{\alpha}(a+b)$

(R2) for $\alpha\neq\pm\beta$: $[x_{\alpha}(a),x_{\beta}(b)]=\prod_{\gamma=i\alpha+j\beta}x_{\gamma}(C^{\alpha\beta}_{ij}a^ib^j)$

(R3) $h_{\alpha}(t)h_{\alpha}(u)=h_{\alpha}(tu)$

where $h_{\alpha}(t):=n_{\alpha}(t)n_{\alpha}(-1)$ and $n_{\alpha}(t):=x_{\alpha}(t)x_{-\alpha}(t^{-1})x_{\alpha}(t)$, and where we replace the (empty) relations (R2) by

(R2') $n_{\alpha}(t) x_{\alpha}(a)n_{\alpha}(t)^{-1}=x_{-\alpha}(t^{-2}a)$

in case $\Phi$ is of type $A_1$.

Question: If $R$ is a Bezout domain with field of fractions $K$, is it known whether the natural map $G_{\Phi}(R)\to G_{\Phi}(K)$ is injective? If not, for which Bezout domains is it known (the answer may then depend on $\Phi$)?

Rings $R$ satisfying this property are called universal in [2] (this property is also sometimes formulated as "$K_2(\Phi,R)$ being generated by Steinberg symbols"). As mentioned in [2, p. 453], fields, $\mathbb Z$ and $k[X]$ for $k$ a field are universal, and it is shown in [2] that $\mathbb Z$ remains universal if we invert suitable finite sets of primes. But I cannot find a reference for arbitrary Bezout domains.

[1] R. W. Carter, Simple groups of Lie type. Pure and Applied Mathematics, vol. 28. John Wiley & Sons, London-New York-Sydney, 1972.

[2] Eiichi Abe and Jun Morita, Some Tits systems with affine Weyl groups in Chevalley groups over Dedekind domains, J. Algebra 115 (1988), no. 2, 450–465.

Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 and p. 198]: it has generators the symbols $x_{\alpha}(a)$ ($\alpha\in\Phi$, $a\in R$) and the following defining relations, where $\alpha,\beta\in\Phi$, $a,b\in R$ and $t,u\in R^{\times}$:

(R1) $x_{\alpha}(a) x_{\alpha}(b)=x_{\alpha}(a+b)$

(R2) for $\alpha\neq\pm\beta$: $[x_{\alpha}(a),x_{\beta}(b)]=\prod_{\gamma=i\alpha+j\beta}x_{\gamma}(C^{\alpha\beta}_{ij}a^ib^j)$

(R3) $h_{\alpha}(t)h_{\alpha}(u)=h_{\alpha}(tu)$

where $h_{\alpha}(t):=n_{\alpha}(t)n_{\alpha}(-1)$ and $n_{\alpha}(t):=x_{\alpha}(t)x_{-\alpha}(t^{-1})x_{\alpha}(t)$, and where we replace the (empty) relations (R2) by

(R2') $n_{\alpha}(t) x_{\alpha}(a)n_{\alpha}(t)^{-1}=x_{-\alpha}(t^{-2}a)$

in case $\Phi$ is of type $A_1$.

Question: If $R$ is a Bezout domain with field of fractions $K$, is it known whether the natural map $G_{\Phi}(R)\to G_{\Phi}(K)$ is injective? If not, for which classes of Bezout domains (e.g. local, PID, Euclidean,...) is it known (the answer may then depend on $\Phi$)?

Rings $R$ satisfying this property are called universal in [2] (this property is also sometimes formulated as "$K_2(\Phi,R)$ being generated by Steinberg symbols"). As mentioned in [2, p. 453], fields, $\mathbb Z$ and $k[X]$ for $k$ a field are universal, and it is shown in [2] that $\mathbb Z$ remains universal if we invert suitable finite sets of primes. But I cannot find a reference for more general Bezout domains.

[1] R. W. Carter, Simple groups of Lie type. Pure and Applied Mathematics, vol. 28. John Wiley & Sons, London-New York-Sydney, 1972.

[2] Eiichi Abe and Jun Morita, Some Tits systems with affine Weyl groups in Chevalley groups over Dedekind domains, J. Algebra 115 (1988), no. 2, 450–465.