On one property of normal subgroups of UT∞(R)

We examine the group of infinite unitriangular matrices. We show that to find a normal subgroup of UT∞(R) for any unital ring R, it suffices to examine properties of matrices in this group having nonzero entries only on some first diagonals. We use this result to find the commutator subgroup of the group of all triangular matrices. We also describe the commutator subgroup of the Vershik–Kerov group.