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

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?

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## 1 Answer

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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hey, thanx, but is there any other way of proving it. I am reading semi simple lie algebras and representations by cahn, and have not covered sl2 theory. –  ramanujan_dirac Mar 6 '12 at 16:58
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