Matricial form of the killing form of $\mathfrak{sl}(n)$

Question

I was reading some of my old notes and I found an explicit matricial form for the killing form $$ \left(\kappa_{ij}\right)=\left(\begin{array}{cccccccc} 4n\\ & \ddots\\ & & 4n\\ & & & 0 & 2n\\ & & & 2n & 0\\ & & & & & \ddots\\ & & & & & & 0 & 2n\\ & & & & & & 2n & 0 \end{array}\right), $$ where the number of $4n$ equals the dimension of the cartan sub-algebra. I'm wondering if there are other explicit form for the killing form and why I don't see them written in every textbook if there are?

Answer 1

Bourbaki's Lie groups and Lie algebras contains, at the end of the 8^th chapter, a table concerning the classical Lie algebras (the four families $A_n$, $B_n$, $C_n$, and $D_n$), wich contains explicit descriptions of the restriction (as a bilinear form) of the Killing form to a Cartan subalgebra.