In this paper, for \(0\le \alpha <Q\) , we consider the operators \(X_{j}(-\Delta )^{-\frac{1+\alpha }{2}},\,j=1,\dots ,n\) , where \(\{\textrm{X}_{j}\}_{1 \le j \le n}\) is a basis of the left-invariant vector fields of degree one on stratified Lie groups \(\mathcal {G}\) and \(\Delta =\sum _{j=1}^{n} \textrm{X}_{j}^{2}\) is the sub-Laplacian of \(\mathcal {G}\) . Firstly, we establish the uniform bound of \(X_{j}(-\Delta )^{-\frac{1+\alpha }{2}}\) on \(\mathcal {G}\) for \(j=1,\dots ,n\) . Secondly, we characterise the uniform boundedness of the commutator \([b,X_{j}(-\Delta )^{-\frac{1+\alpha }{2}}]\) via \(\textrm{BMO}(\mathcal {G})\) space for \(j=1,\dots ,n\) . Thirdly, we give the characterisation of the uniform compactness of \([b,X_{j}(-\Delta )^{-\frac{1+\alpha }{2}}]\) with respect to \(\textrm{VMO}(\mathcal {G})\) space for \(j=1,\dots ,n\) .
