Hi all 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?