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so i promised you in the last video id explain why the weight lattice of su 3 is this triangular lattice um with sort of angles of 120 or 60 degrees and the reason is these weight losses actually have an intrinsic notion of geometry they know about angles and lengths and things and that geometry comes from a structure called the killing form so let little g be a lay algebra the killing form of g is the map k from g times g to the field were working over which most of the times c um defined by k of x comma y equals the trace of add x times add y so what does this mean add x remember is the thing that sends zed in the lia algebra to x bracket z so add x is a map from the algebra to itself add y is a map from the algebra to itself add x add y is the map from little g to little g that sends z to x bracket y bracket z okay thats a linear map from the algebra to itself we can think of it as a matrix if we pick a basis of g and then we take the trace of that matrix so i said pick a basis
