Abstract
We define subalgebras <TEX>${V_q}^{mZ{\times}nZ}\;of\;V_q\;where\;V_q$</TEX> are in the paper [4]. We show that the Lie algebra <TEX>${V_q}^{mZ{\times}nZ}$</TEX> 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 <TEX>${V_q}^{mZ{\times}nZ}$</TEX> in this paper.
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