TITLE: Question about 8.4 in Humphreys
QUESTION [8 upvotes]: I am reading section 8.4 in Humphreys' book Introduction to Lie Algebras and Representation Theory. He is showing that the only scalar multiples of a root are 1 and -1, but I have trouble understanding his reasoning: He considers the direct sum $M=\bigoplus_{c\in\mathbb{F}} L_{c\alpha}$ for some fixed root $\alpha$. He shows that the only even weights of $h_\alpha$ on $M$ are 0, 2 and -2. Then, he says that this proves that twice a root is never a root. Why is that true? The function $\frac 12\alpha$ could still be a root.

REPLY [4 votes]: If $\alpha$ and $\beta=\alpha/2$ were both roots, then we would have a situation, where $2\beta$ and $\beta$ are both roots. Thus $h_\beta$ would have even weights $0,\pm2,\pm4$ on $M=\bigoplus_{c\in F}L_{c\beta}$ contradicting  the fact just proven.