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Let $L$ be a semisimple Lie algebra, ad let $\Phi$ be a root system. Fix a fundamental root system $\Delta$ of $\Phi$ with corresponding to $\Phi^+$.

I would like to understand the subalgebra generated by all $L_{\alpha}$ and $L_{-\alpha}$ where $\alpha\in \Omega\subset\Delta$. Is this subalgebra semisipmle? Can we say something about its Dynkin diagram?

Ronald
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    The Dynkin diagram says everything. Start with the diagram for the lie algebra $L$, remove the nodes in $\Delta\backslash\Omega$. The resulting diagram is the diagram for your new Lie algebra. – David Hill Apr 08 '16 at 20:26

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