Commutator subgroup of Vershik–Kerov group

We describe a commutator subgroup of Vershik–Kerov group over an infinite field and find the bound for its commutator width. This gives a partial solution of the problem posed by Sushchanskii in 2010. We also describe the lower central series of the group of infinite upper triangular matrices over an infinite field and find the bound for its commutator width.