  If $A$ is a unital associative ring and $\ell \geq 2$, then the general
linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_\alpha$ and Weyl
elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell
- 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl
elements for $\ell \geq 4$ satisfying natural conditions, then there is a way
to recover the ring $A$. We prove a generalization of this result not using the
Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct
a non-unital associative ring with a well-behaved Peirce decomposition.
