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?
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.
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