Corollary 3

Suppose that $K$ is algebraically closed. If $g$ is an $r$-dimensional solvable Lie algebra, every ideal of $g$ is a term of a decreasing sequence of ideals of dimensions $r, r-1, \ldots, 0$.

Every ideal is part of a Jordan-Hölder series of $g$, considered as the space of the adjoint representation (Algebra, Chapter I, § 6, no. 14, Corollary to Theorem 8); then it suffices to apply Corollary 2.