Dimension of the Lie algebra of $SO(n,k)$

Question

Let $SO(n,k)$ denote the special generalized orthogonal group. Of course, $SO(n,k)$ is a Lie group. I know that the Lie algebra $so(n,k)$ of $SO(n,k)$ coincides with the Lie algebra of $O(n,k)$. I'd like to know the dimension of $so(n,k)$. I do not need a proof, I'm just curious about it.

Answer 1

I assume that that notation $so(n, k)$ means signature of a quadratic form (it also could be dimension and basic field, and quadratic form is a standard form ...)

Dimension of $so(Q)$ does not depend on the quadratic form $Q$, in fact, for any form there is explicit isomorphism of $so(Q)$ and space of skew-symmetric matrices (only as vector spaces!). Therefore, for any signature of a quadratic form dimension is the same $(n+k)(n+k-1)/2$.

I don't want to copy this construction here, but it is explained in the Fulton and Harris book "Representation theory. First course" in the section 20.1, in my edition it is formula 20.4.