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Suppose the Vogan diagram of $E_6$ is given where the root $\alpha_2$ is painted. Now from this diagram how shall I compute the $k_0$ part of $g_0$, Where $g_0 = k_0 + p_0$ is the Crartan decomposition?

It is given in the book "Lie Groups Beyond an Introduction by A W Knapp" in the page 416. the answer is $su(6)+su(2)$ but i did not understand the argument in page 417. Please help me.

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This question is better suited for math.stackexchange.com –  MTS May 29 '13 at 0:39
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