Let $V$ be finite dimensional vector spaces and $q$ is quadratic form. I'm looking for $Z(SO(V,q))$. where $SO(V,q)$ is special orthogonal group.
If $\operatorname{dim} V$ is odd then $Z(SO(V,q))={I}$ because $-\sigma_u\in SO(V,q) $ where $\sigma_u$ is reflection $\sigma_u(x)=x-2\frac{b(x,u)}{q(u)}$ But I couldn't calculate center when $\operatorname{dim} V$ is even.
I have not any idea how to deal with it.
Thanks.