I have a question about the computation of positive roots/coroots and Borel subgroups of the indefinite group $G = \operatorname{GSO}(n, 2)$ for some $n \in \mathbb{Z}^{+}$, where the form defining the group $G$ is $$ \begin{pmatrix} 1_{n} & 0 \\ 0 & -1_{2} \end{pmatrix} \,. $$
Given $\mu \colon z \mapsto \operatorname{diag}(1, 1, \dotsc, 1, z, z^{-1})$, a cocharacter of $G$, I want to write $\mu$ as a sum of positive roots (with respect to some choice of maximal torus $T$ and Borel $B$ of $G$). But I couldn't find an explicit matrix description of $B$ and $T$ in this case. Is there a way to find them?
Even releated with 1, after writing $\mu$ as a sum of positive roots (or even in terms of fundamental weights if we know them explicitly), I want to decide whether $\mu$ is a highest weight vector or not and if it is, then to find the dimension of the corresponding highest weight representation. But $G$ is not a simple group so I guess the standard fundamental module/dimension argument does not valid in this case. Is there a way to compute the highest weight representations/dimensions for $\operatorname{GSO}(n, 2)$? Or even the same question for same kind of non-simple similitude groups like $\operatorname{GSp}$, $\operatorname{GSpin}$, $\operatorname{GU}$, etc. ?