Let (X, d) and \((Y,\rho )\) be compact metric spaces, let \(\tau\) and \(\eta\) be Lipschitz involutions on (X, d) and \((Y,\rho )\) , respectively, and let \(\alpha ,\beta \in (0,1)\) . We first give some sufficient conditions that a map T between real little Lipschitz algebras with Lipschitz involution \({\rm{lip}}(X,d^\alpha ,\tau )\) and \({\rm{lip}}(Y,\rho ^\beta ,\eta )\) be an order isomorphism. We next prove that if \(T:{\rm{lip}}(X,d^\alpha ,\tau )\rightarrow {\rm{lip}}(Y,\rho ^\beta ,\eta )\) is an order isomorphism with \(|T(i{\rm{Im}}\,f)|\le T(|{\rm{Im}}\,f|)\) for all \(f\in {\rm{lip}}(X,d^\alpha ,\tau )\) and \(|T^{-1}(i{\rm{Im}}\,h)|\le T^{-1}(|{\rm{Im}}\,h|)\) for all \(h\in {\rm{lip}}(Y,\rho ^\beta ,\eta )\) , then T is a homeomorphism from \({\rm{lip}}(X,d^\alpha ,\tau )\) with Lipschitz sum norm topology to \({\rm{lip}}(Y,\rho ^\beta ,\eta )\) with Lipschitz sum norm topology, \(T(1_X)\) and \(T^{-1}(1_Y)\) are nonvanishing positive functions on Y and X, respectively, and there exists a bijective map \(\Phi :Y_\eta \rightarrow X_\tau\) such that \(T(f)(y)=T(1_X)(y)f(x)\) for all \(y\in Y\) , \(x\in \Phi (y_\eta )\) and \(f\in {\rm{lip}}(X,d^\alpha ,\tau )\) with \(f(x)\in \mathbb {R}\) , where \(x_\tau =\{x,\tau (x)\}\) for all \(x\in X\) , \(X_\tau =\{x_\tau : x\in X\}\) , \(y_\eta =\{y,\eta (y)\}\) for all \(y\in Y\) and \(Y_\eta =\{y,\eta (y)\}\) . Finally, we prove that every order isomorphism between real little Lipschitz algebras \({\rm{lip}}_{\mathbb {R}}(X,d^\alpha )\) and \({\rm{lip}}_{\mathbb {R}}(Y,\rho ^\beta )\) is an essential weighted composition operator.
