The Jacobi-identity IS THE derivation in all representations:

[A,[B,C]] = [[A,B],C] + [B,[A,C]] , take A = "D" for derivation.

however, what is "GAP code"?

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.

with this outer derivations you get a Lie algebra extension, still a Lie algebra module.

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.

I've now managed to calculate all of the derivations of the 4-dimensional solvable Lie algebras and have found the example I was looking for.