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I found these equalities from Gordon Brown (1963).

He uses the killing form to measure the length of the roots in a semi simple algebra.

First and second equalities are quite obvious and come from the definition.

Could you help me for the last one which prove that we have a projector?

It is Ʃ_g g(h_a) g(h_b )=Ʃ_g (a,g)(g,b)

He writes that the sum of the squared lengths is the dimension of the Cartan subalgebra.

If we take the su(2) weak interaction, we have two ladder bosons W- W+ (corresponding to opposite roots) adding or substracting one unit to isospin and dim H = 1.

What is the relation with the length 1/√2 he finds for each root?

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# Length of roots in su(2) su(3) and other Lie algebras.

Can you offer guidance or do you also need help?

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