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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?

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  • Is $t$ (or all of $T$) contained in $H$? If not, I don't understand what it would mean for a root in $\Phi$ to "take a value" at $t$, positive or not: because by definition the roots are elements of the dual space of $H$, hence can only be evaluated at elements of $H$. – Torsten Schoeneberg May 09 '18 at 22:57
  • If, on the other hand, $t \in H$, then I think it is quite easy to show that with the modified definition one gets a possibly smaller $S'$, which is contained in the original $S$, but certainly also contains $x$ and is a subalgebra (hence solvable, because it's a subalgebra of the original $S$). – Torsten Schoeneberg May 12 '18 at 04:14
  • I agree that there is no reasons to consider $\alpha$'s taking positive rational value rather than positive integeral. (Maybe emphasis on characteristic zero?) (+1) – luxerhia May 06 '20 at 05:28

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