Groups with 𝖠ℓ-commutator relations

If A A is a unital associative ring and ℓ ≥ 2 ≥ 2 , then the general linear group GL ⁡ ( ℓ , A ) GL( , A) has root subgroups U α U_α and Weyl elements n α n_α for α α from the root system of type A ℓ − 1 A- 1 . Conversely, if an arbitrary group has such root subgroups and Weyl elements for ℓ ≥ 4 ≥ 4 satisfying natural conditions, then there is a way to recover the ring A A . A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring M ⁡ ( ℓ , A ) , M( , A), a nonunital associative ring with a well-behaved Peirce decomposition is provided.