In this paper, using some properties about Toeplitz kernels, we present some results about finite-rank properties of the commutator [Af,Ag]\documentclass[12pt]{minimal}
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\begin{document}$$[A_f,~A_g]$$\end{document}. Firstly, we show that [ABn,Av∗]\documentclass[12pt]{minimal}
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\begin{document}$$[A_{B_n},~A_v^*]$$\end{document} must have a finite rank on the model space Ku2\documentclass[12pt]{minimal}
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\begin{document}$$K_u^2$$\end{document}, where Bn\documentclass[12pt]{minimal}
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\begin{document}$$B_n$$\end{document} is a finite Blaschke product and v is an inner function. Next, we present that when kerTu¯Bn\documentclass[12pt]{minimal}
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\begin{document}$$\text {ker}~T_{\overline{u}B_n}$$\end{document} is an invariant subspace of Tϕ∗\documentclass[12pt]{minimal}
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\begin{document}$$T_\phi ^*$$\end{document}, then [ABn,Aϕ∗]\documentclass[12pt]{minimal}
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\begin{document}$$[A_{B_n},~A_\phi ^*]$$\end{document} has a finite rank on Ku2\documentclass[12pt]{minimal}
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\begin{document}$$K_u^2$$\end{document} for ϕ∈H∞\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in H^\infty $$\end{document}. Finally, we prove that [ABn,Aϕ∗]\documentclass[12pt]{minimal}
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\begin{document}$$[A_{B_n},~A_\phi ^*]$$\end{document} must have a finite rank on Ku2\documentclass[12pt]{minimal}
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\begin{document}$$K_u^2$$\end{document} when u=Bnu1\documentclass[12pt]{minimal}
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\begin{document}$$u=B_nu_1$$\end{document} for an inner function u1\documentclass[12pt]{minimal}
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\begin{document}$$u_1$$\end{document}.
