Let $L$ be a semisimple Lie algebra over a field $F$ (which is algebraically closed with characteristic 0) and $\alpha$ and $c \alpha$ for $c \in F$ are roots of $L$. I want to prove that $c \in \{-1, 1\}$.
I know the argument for the statement: $\alpha$ is a root $\implies$ $-\alpha$ is a root. But I have not seen the argument for the above statement which means that the only possible multiples are $-\alpha$ and $\alpha$. I have looked here on StackExchange and in "Introduction to Lie algebras and representation theory" from Humphreys but have not found anything.
