I'm trying to get my hands on the *general spin group* $G = \textrm{GSpin}_{2n}$. It have seen it mentioned as a connected, reductive group whose derived group is $\textrm{Spin}_{2n}$, which is the unique semisimple, simply connected group having the same root system as $\textrm{SO}_{2n}$.

The general spin group is for example mentioned in a paper here https://math.okstate.edu/people/asgari/Files/gspin.pdf .

I have read that it is difficult to view $\textrm{Spin}_{2n}$ as a closed subgroup of some general linear group. According to (9.16) in *Linear Algebraic Groups and Finite Groups of Lie Type*, the smallest $m$ for which $\textrm{Spin}_{2n}$ can be embedded in $\textrm{GL}_{m}$ is $2^{[\frac{2n-1}{2}]}$.

As for $G$, I am not really sure how that should be defined. Since its derived group is $\textrm{Spin}_{2n}$, it should be quotient of $\textrm{Spin}_{2n} \times S$ by a finite normal subgroup, where $S$ is some torus.

So my questions are:

1 . How should the general spin group be defined?

2 . What is the most straightforward way to compute the roots, coroots, and root datum of the general spin group as well as its derived group?

For the group $\textrm{SO}_{2n}$, I did (2) by finding a maximal torus and computing the Lie algebra. I imagine there must be a different approach when one is not working with an explicit embedding into $\textrm{GL}_{m}$.

evenorthogonal groups here? The group SO$_{2n+1}$ and its simply connected covering group Spin${2n+1}$ (of Lie type $B_n$) are comparable in many respects to the groups of type $D_n$. . $\endgroup$