Exercise 1, § 7

(a) For a Lie $K$-algebra to be nilpotent, it is necessary and sufficient that it be isomorphic to a subalgebra of an algebra $\mathfrak{n}(n, K)$.

(b) Suppose that $K$ is algebraically closed. For a Lie $K$-algebra to be solvable, it is necessary and sufficient that it be isomorphic to a subalgebra of an algebra $\mathfrak{t}(n, K)$.