How to choose a basis for the diagonal matrices?

Tutor's Answer

(Top Tutor) Studyfaq Tutor
We choose a basis for the diagonal matrices as follows: denote E_{i,j} as the matrix with a 1 in the i,j’th position and 0’s everywhere else. For diagonal matrices the basis is h_i=E_{i,i}-E_{i+1,i+1}. Now we want to calculate the Killing form for these matrices, so we choose a diagonal matrix H, with diagonal entries e_i(H) for some set of diagonal entries. The commutator [H,E_{i,j}]=(e_i(H)-e_j(H))E_{i,j}, shows us that the Tr(ad_had_h) can be written as the following sum: [k(H,H)=sum_{i,j}(e_i(H)-e_j(H))^2]which we can expand the...
Completed Work