A Lie algebra g over a field F is a vector space together with a bilinear
map [ , ] satisfying [x ,x ] = 0 in addition to Jacobi identity . A Lie subalgebra
B of a Lie algebra g is said to be a Cartan subalgebra if it is a nilpotent and
equals its
normalizer, and it is proved that semi simple Lie algebra g
decomposes into weight spaces for B.
In this scientific paper we present the conception of distinguished
element 0 h in finite dimensional semi simple Lie algebra over a field F has
characteristic 0 and we will prove that the previous decomposition g into
weight spaces for B is the same to decomposition g as a direct sum of h0 ad eigen
spaces. This leads us to construct algorithm to test simple Lie algebras.
We programmed the previous algorithm to test simple linear Lie algebras
over a numeral field by Mathematica 5.0 program where applied this algorithm
on semi simple linear Lie algebra SL(3, ) to prove that it is simple.
