Exercise 2, § 5

(a) Show that in the 3-dimensional solvable Lie algebra over $\mathbf{R}$ defined by the multiplication table $[x, y] = z,\ [x, z] = -y,\ [y, z] = 0$, there exists no decreasing sequence of ideals of dimensions 3, 2, 1, 0.

(b) Show that in the non-commutative 2-dimensional solvable Lie algebra $g$ there exists a sequence of ideals of dimensions 2, 1, 0 but that $g$ is not nilpotent.