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I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) involves showing that $[\frak g_{\alpha},g_{\beta}] = g_{\alpha+\beta}$ where $\alpha$, $\beta$, and $\alpha+\beta$ are roots of the semisimple Lie algebra $\frak g$. The book states that it is sufficient to show that $[\frak g_{\alpha}, g_{\beta}] \scr \ne 0$. Why is this sufficient? Is it because $\frak g$ is semisimple? I'm not sure why it is obvious from that though.

Also, any hints on how to do this would be appreciated. So far, I've shown that $[\frak g_{\alpha},h] \scr \ne 0$ where $\frak h$ is the Cartan subalgebra of $\frak g$, but I don't know how to connect that back to 2 roots. I was trying to do something algebraic involving Jacobi's identity, but haven't had any success so far. Thanks for any help!

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DO you know at that point that the root spaces are $1$-dimensional? – Mariano Suárez-Alvarez Dec 14 '11 at 2:01
Yes, the root spaces are 1-dimensional. Does the bracket being nonempty automatically imply that it must be a 1-dimensional space (and hence all of $\frak g_{\alpha+\beta}$)? – Misha Dec 14 '11 at 2:21
$[g_\alpha,g_\beta]$ is a subspace of $g_{\alpha+\beta}$; if the latter is $1$-dimensional, the former can only be $0$-dimensional and $1$-dimensional. In the second case, it in fact must equal $g_{\alpha+\beta}$. – Mariano Suárez-Alvarez Dec 14 '11 at 2:26
That makes sense, thank you very much Mariano! Any suggestions on how to show that the bracket is in fact nonempty? – Misha Dec 14 '11 at 2:29

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