If $\alpha$ is a root of a simple lie algebra, then prove that the only multiples of $\alpha$ which are roots are $\alpha, -\alpha,0$

Question

If $\alpha$ is a root, then the only multiples of α which are roots are $\alpha, -\alpha, 0$. Here $\alpha$ is a root of a simple lie algebra. How do I prove this?

Answer 1

An argument showing this is given on page 38 of Humphreys' Introduction to Lie Algebras and Representation Theory, #9 in Springer GTM series. It is based on knowledge of $sl_2$-theory and some other basic properties.