The proposition is from Humphreys.
I don't understand how to prove the highlighted statements. How can I express a general element of K? I tried using Cartan decomposition of L but it doesn't work.
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There is nothing to prove. We have $[K,L_{\beta}]=0$ by the definition of $K$ and because of $[L_{\alpha},L_{\beta}]\subseteq L_{\alpha+\beta}=0$. For the meaning of "centralized" and "normalized" see here. – Dietrich Burde Jan 27 '15 at 21:17
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My problem is the meaning of "subalgebra generated by". Is it the direct sum of the generators? – Ginevra Carbone Jan 27 '15 at 21:41
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@magnetissimo The subalgebra generated by a set of elements $A$ is the smallest subalgebra containing $A$. A subalgebra is a subspace closed under the bracket. – Matt Samuel Jan 27 '15 at 22:10