Steinberg relations

Remarks

Example: the elementary matrices $e_{ij}(r) = I + r E_{ij}$ in $GL_n(R)$ satisfy the Steinberg relations, where $E_{ij}$ is the matrix with $1$ at position $(i, j)$ and $0$ elsewhere. Indeed $E_{ij} E_{kl}$ equals $E_{il}$ if $j = k$ and $0$ otherwise. So $(I + rE_{ij})(I + sE_{ij}) = I + (r + s)E_{ij}$, because $E_{ij}^2 = 0$ for $i \ne j$. For pairwise distinct $i, j, l$, a direct product of the four factors gives $[e_{ij}(r), e_{jl}(s)] = I + rs E_{il}$. If $j \ne k$ and $i \ne l$, then $E_{ij}E_{kl} = 0 = E_{kl}E_{ij}$, so the two matrices commute.

Non-example: the case $l = i$ is excluded in clause 2. For $n = 2$ and $R = \Z$, the matrices $e_{12}(1)$ and $e_{21}(1)$ have commutator $\begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}$, which is not an elementary matrix. So no relation of the form of clause 2 holds for the pair $(i, j, l) = (1, 2, 1)$.

Clause 1 implies $x_{ij}(0) = 1$ and $x_{ij}(r)^{-1} = x_{ij}(-r)$.