  In this paper, the main aim is to consider the boundedness of the fractional
maximal 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 of
the Lipschitz spaces on Lie group are given.
