If a Borel subalgebra $\mathfrak b$ of a semisimple Lie algebra $\mathfrak g$ contains a Cartan subalgebra $\mathfrak h$, then we have a root space decomposition $$ \mathfrak b = \mathfrak h \oplus \left(\oplus_{\alpha \in S} \mathfrak g_{\alpha}\right) $$ with $S$ being some subset of the set $R$ of roots of $\mathfrak g$. The fact that $\mathfrak b$ is solvable already asserts that $S$ cannot contain both $\alpha$ and $-\alpha$ for any $\alpha \in R$.
From this, can we conclude that $S = R^+$ with respect to some basis of $R$?
Humphreys (Chapter 16) states this result as a consequence of the conjugacy of the Cartan subalgebras and the Borel subalgebras, but it feels like this should be a standalone result that can be concluded from basic properties of root systems and the fact that the Borel subalgebras are maximally solvable.