Exercise 2, § 10

Let $m=\frac{1}{2}$(dim($\mathfrak{g}$) $+$ rk($\mathfrak{g}$)). Every solvable subalgebra of $\mathfrak{g}$ is of dimension $\leq m$; if it is of dimension $m$, it is a Borel subalgebra. (Reduce to the algebraically closed case, and use Th. 2.)