Abelian ideal of 8-dimensional nilpotent Lie algebra

Question

Let $$L : [x_1, x_2] = [x_3, x_4] = x_6, [x_1, x_5] = [x_2, x_3] = x_7, [x_1, x_7] = [x_2, x_4] = [x_4, x_5] = [x_6, x_3] = x_8$$ be a nilpotent Lie algebra of dimension $8$. Does have an abelian ideal? I guess this Lie algebra has no abelian ideal of dimension 5 but I can' t show it.

Answer 1

The subspace of basis $(x_2,x_5,x_6,x_7,x_8)$ is clearly an ideal (since it contains the derived subalgebra). Moreover it's abelian (since all the given nonzero brackets involve one of $x_1,x_3,x_4$).