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In this paper, the main aim is to consider the boundedness of the fractionalmaximal commutator $M_{\alpha,b}$ and the nonlinear commutator $[b,M_{\alpha}]$ on the Lebesgue spaces over some stratified Lie group $\mathbb{G}$when $b$ belongs to the Lipschitz space, by which some new characterizations ofthe Lipschitz spaces on Lie group are given.

Some estimates for commutators of the fractional maximal function on stratified Lie groups