I came across a small question while doing proof of conjugacy of Cartan subalgebras from Humphreys' Lie algebra; Page 85-86.
Let $L$ be semisimple, finite dimensional over $\mathbb{C}$; $H$ maximal toral and $\Phi$ the relative root system.
Let $B$ be a mxaimal solvable subalgebra (Borel) in $L$ and $T$ a toral (may be non-maximal) in $B$.
Suppose $T$ acts non-trivially on $B$ via adjoint. So there exists a common eigenvector for $T$ in $B$ say $x$ such that it is not killed by some $t\in T$ (under adjoint action). Let $[t,x]=cx$ where $c\neq 0$. Replacing $t$ by $\frac{1}{c}t$ we may assume that $c=1$ (positive integral). Define $$S:=H\oplus (\oplus_{\alpha}L_{\alpha} )$$ where $\alpha$ runs over those roots in $\Phi$ which take positive rational value at $t$.
For this $S$, it was then proved that $S$ is (subalgebra)+(solvable)+(contains $x$).
Q. In definition of $S$, why $\alpha$'s in $\Phi$ are considered with $\alpha(t)\in\mathbb{Q}_{\geq 0}$ than $\alpha(t)\in\mathbb{Z}_{\geq 0}$? Where this modification affects in the three properties above of $S$ written in brackets?