Killing Form of Orthogonal Lie Algebra

Theorem
Let $\mathbb K\in\{\C,\R\}$.

Let $n$ be a positive integer.

Let $\mathfrak{so}_n(\mathbb K)$ be the Lie algebra of the special orthogonal group $\operatorname{SO}_n(\mathbb K)$.

Then its Killing form is $B : (X,Y) \mapsto (n-2)\operatorname{tr}(XY)$.

Lemma
Let $R$ be a ring with unity.

Let $n$ be a positive integer.

Let $E_{ij}$ denote the matrix with only zeroes except a $1$ at the $(i,j)$th position.

Then for all $X,Y\in R^{n\times n}$:
 * $\sum_{1\leq i < j \leq n}\operatorname{tr}\left( (X (E_{ij}-E_{ji}) Y)^t (E_{ij}-E_{ji}) \right) = \operatorname{tr}(Y)\operatorname{tr}(X) - \operatorname{tr}(Y^t X)$

Proof
Use Trace of Alternating Product of Matrices and Almost Zero Matrices.

Use Definition:Frobenius Inner Product and Trace in Terms of Orthonormal Basis and the fact that the $(E_{ij}-E_{ji})/\sqrt2$ are an orthonormal basis of $\mathfrak{so}_n$.

Also see

 * Killing Form of Symplectic Lie Algebra