Stable elementary subgroup ($E_{\infty }(A)=⟨I+a{e}_{ij}:i\ne j⟩$)

The subgroup generated by elementary transvections in finite matrix sizes, with identity stabilization. A third distinct index expresses each transvection as a commutator; stable block identities show that this subgroup is normal and that its quotient in the stable general linear group is abelian. Thus it is precisely the commutator subgroup.