Killing Form of Orthogonal Lie Algebra

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

Let $n$ be a positive integer.

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

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

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

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 $\left({E_{ij} - E_{ji} }\right) / \sqrt 2$ are an orthonormal basis of $\mathfrak {so}_n$.

Also see

 * Killing Form of Symplectic Lie Algebra