$\mathfrak so(V,B)$ as subalgebra and trace if subsets of it.

Question

I'm studying lie algebras, and got stuck on this one:

Let $B$ be a bilinear form on a finite-dimensional vector space $V$ over $\mathbb F$.

I've seen many books that say that $\mathfrak so(V,B)$ is a lie subalgebra of $\mathfrak gl(V)$, but didn't find a proof.

and also, in the case that B is non-degenerate, how can I show that $tr(X)=0$ for all $X \in \mathfrak so(V,B)$ ?

Thank you.