中文摘要：

给出有限维Lie代数Killing型的计算方法和机械化实现，本算法适合于任意有限维Lie代数，适用于非线性偏微分方程对称性的判断和应用．以此算法为基础，给出了1＋2维标准双曲和抛物方程拥有Lie代数性质的判定，得到该两个经典方程的Lie代数结构及相关性质．给出了1＋2维抛物方程Lie代数的优化系统，为从代数角度研究偏微分方程问题提供了可借鉴的结果．

英文摘要：

A mechanical algorithm for Killing form of a finite dimensional Lie algebra is given. This algorithm can be used to decide the symmetry properties of nonlinear partial differential equa- tions （PDEs）. Based on the algorithm, the structures of the Lie algebras of the standard 1 ＋ 2 dimensional hyperbolic and parabolic PDEs and some algebraic properties of them are presented. The optimal system for a 1 ＋2 parabolic PDE has been shown to generalize the existing literature results on 1 ＋ 1 one. These provide an alternative ways to study PDEs from the algebraic points.