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I am reading books on elementary particle and i see that their

gauge bosons may be neutral or have opposite charge. They live

in semisimple Lie algebras. So I searched in math books how to prove

that in a semisimple Lie algebra if α is a root so is -α.

I found that it is related to the fact that the killing form is not degenerate.

Could you comment this:

If α is a root the space g^{α}is not nul and orthogonal to Ʃ_{λ ≠- α}g^{ λ}. Since the Killing form is non degenerate g^{-α}must be ≠ 0 then -α is a root.

Here g^{λ}= {x ∈ g |∃n ∈ N ∀h ∈ h , (π(h) − λ(h))^{n}(x) = 0}.

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# Proof of existence of opposite roots in semisimple algebras?

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