Groups with $\mathsf A_\ell$-commutator relations

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.