 Killing form of the special linear Lie algebra

On the complex special linear Lie algebra ${\mathfrak{sl}}_n(\mathbb{C})$, the Killing form isThus its intrinsic definition through the Adjoint representation becomes an ordinary matrix trace calculation.

To derive the coefficient, act first on the full space of $n\times n$ matrices. Write $L_X(A)=XA$ and $R_X(A)=AX$. The Adjoint representation acts by ${\mathrm{ad}}_X=L_X-R_X$. In the basis of matrix units, direct calculation givesHere $\mathrm{Tr}$ is the trace on the $n^2$-dimensional space of matrices. Expanding the product yieldsFor traceless $X,Y$, the last term vanishes. The full matrix space splits as ${\mathfrak{sl}}_n(\mathbb{C})\oplus \mathbb{C}I$, and every commutator acts trivially on the scalar summand. Removing that summand therefore leaves the trace unchanged, giving the Killing form on ${\mathfrak{sl}}_n(\mathbb{C})$.