Killing Form of Orthogonal Lie Algebra

Theorem

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

Let $n$ be a positive integer.

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

Then its Killing form is $B: \tuple {X, Y} \mapsto \paren {n - 2} \map \tr {X Y}$.

Proof

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 $\tuple {i, j}$th position.

Then for all $X, Y \in R^{n \times n}$:

$\ds \sum_{1 \mathop \le i \mathop < j \le n} \map \tr {\paren {\map X {E_{ij} - E_{ji} } Y}^t \paren {E_{ij} - E_{ji} } } = \map \tr Y \map \tr X - \map \tr {Y^t X}$

$\Box$

This theorem requires a proof. In particular: 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 $\paren {E_{ij} - E_{ji} } / \sqrt 2$ are an orthonormal basis of $\mathfrak {so}_n$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Killing Form of Symplectic Lie Algebra