We define subalgebras ${V_q}^{mZ{ imes}nZ};of;V_q;where;V_q$ are in the paper [4]. We show that the Lie algebra ${V_q}^{mZ{ imes}nZ}$ is simple and maximally abelian decomposing. We may define a Lie algebra is maximally abelian decomposing, if it has a maximally abelian decomposition of it. The F-algebra automorphism group of the Laurent extension of the quantum plane is found in the paper [4], so we find the Lie automorphism group of ${V_q}^{mZ{ imes}nZ}$ in this paper.