I'm not sure what you mean Anton. A derivation D of a Lie algebra L is a linear transformation from L to L such that D([x,y]) = [x,D(y)] + [D(x),y]. So, by the Jacobi identity, the adjoint map (left multiplication) is a derivation. However, in general, there are others (outer derivations). GAP is an algebraic manipulation package.
Ah, I see what you mean Anton. I'm trying to determine whether the nilradical N(L) is stable under all derivations of the algebra L when L is solvable. I don't see how considering the extension helps with this.