OK, firstly I hope this is the rigth place for my question. I'm in a bit of a problem. I need to be able to calucalte the Killing for for a Lie algebra by next wek, but I'm stuck and won't be able to get any help in 'real life' until Friday, not leaving me enough time to sort out my problem. So I was hoping someone here might be able to show me some pointers. So I have the Lie algebra of all upper triangular 2x2 matrices and am using the basis (1 0) (0 1) (0 0) (0 0),(0 0),(0 1). (I hope these matrices turn out OK: they should be 3 2x2 matrices.) The Killing form is defined as K(X,Y) = trace(adX adY) for all X,Y in the Lie algebra, where ad is the adjoint, defined on Z by adX(Z) = [X, Z] = XZ-ZX. I know I have to consider adX and adY as linear transformations and work out their matrices (and hence multiply them together and find the trace). What I'm not sure is how to go about this, and at what point do I use the bases instead of a general element from the Lie algebra? Any help would be much appreciated, as I've spent a fair bit of time on this and got no where. I also have to so the same for the special linear Lie algebra sl_2(C) (C=complex nubers), but I think if I have some hints for the triangular one I should hopefully be able to work the special linear one out my self.