The Classification Of 2-dimensional Lie Triple Systems

In this paper, We study the classification and the properties of their derivation algebras of 2-dimensional Lie triple systems over the complex(real) number field .And we prove them using the ternary operation [a, b, c] of Lie triple system.In section 1 ,we give the basic definitions and some basic properties of Lie Triple System. Such as definitions of subalgebras, ideals, solvability, solvable ideals, radical, derivations..Secondly, we recall the relation between the multiplication of 2-dimensional Lie triple system and symmetric bilinear functions, then using it to classify the 2-dimensional Lie triple system over the complex(real) field and discuss their solvability.At last, we get the derivation algebras of Lie triple system and matrix forms.