What is the Killing form of SO(3,1)?

Question

I computed the Killing form of SO(3,1). Now I would like to check the correctness of the result but I could not find this Killing form in any publication. Where can I find it?

Answer 1

We can find it in books on Lie algebras:

Proposition: The Killing form of $\mathfrak{so}(p,q)$ is given by $B(X,Y)=(q+p−2)tr(XY)$.

A possible reference is this book. The result can be obtained in a similar way as in in this post:

Killing-form $\mathrm{Tr}(\mathrm{ad}_{X} \circ \mathrm{ad}_{Y})$ of $\mathfrak{so}(n)$. The Killing form in these cases is always proportional to the trace form.