Classification And Dimension Of B-subalgebras Of S(n)

S(n) is atypical finite dimensional restricted simple Lie algebras with over an algebraically closed field with positive characteristic.It is very important and difficult to classify all maximal solvable subalgebras.In this paper,we just focus on maximal completely solvable subalgebras which is called B-subalgebra.So far,the conjugated classifications of maximal completely solvable subalgebras of W(n)-type Lie algebra have been solved.However,there is no substantial progress in the classification of H(2r),S(n)and K(2r+1).This paper will describe the conjugated classification of B-subalgebra in S(2).In order to obtain the B-subalgebra in S(2),we will restrict the B-subalgebra in W(2)to S(2).In the end,we find two non-conjugated B-subalgebras under the general situation and calculate the graded dimension of two B-subalgebras The ideas and methods in this article are instructive for the study of the general situation of S(n)-type.