TITLE: Definition of $\textrm{GSpin}_{2n}$ and its root datum
QUESTION [5 upvotes]: 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 $\textrm{GL}_{m}$.

REPLY [2 votes]: If you'd like to be completely explicit, the Atlas of Lie Groups and Representations software can do this. See www.liegroups.org. Comments are in braces { }.
atlas> {start with simply connected group Spin(8)xGL(1),
of type D4.T1, then mod out by specified element of order 2
so G=Spin(8)xGL(1)/Z_2
Z_2 given by the vector [1/2,1/2,1/2]: element of center
of Spin(8)xGL(1)=Z/2 x Z/2 x C^x, i.e. the element (-1,-1,-1)}
atlas> set rd=root_datum(Lie_type ("D4.T1"),[1/2,1/2,1/2])
Variable rd: RootDatum
atlas> rd
Value: simply connected root datum of Lie type 'D4.T1'
{derived group is simply connected}
atlas> set roots=simple_roots (rd)  
{simple roots as columns of a matrix}
Variable roots: mat
{The software chooses a basis so that X^*(T)=Z^5, X_*(T)=Z^5, 
with the dot product pairing, and then gives roots and corooots 
as integral vectors}
atlas> set coroots=simple_coroots (rd)  {simple coroots as columns}
Variable coroots: mat
atlas> roots
Value:  {roots are columns, 4 vectors of size 5} 
|  2, -1,  0,  0 |
|  3,  0, -1, -1 |
|  1,  0,  0,  0 |
| -2,  0,  0,  2 |
|  0,  0,  0,  0 |

atlas> coroots
Value: {coroots are columns}
| 1, -2,  1, 1 |
| 0,  1, -2, 0 |
| 0,  0,  2, 0 |
| 0,  0, -1, 1 | 
| 0,  0, -1, 0 |

atlas> ^roots*coroots  {compute the pairing of roots and coroots}
Value: 
|  2, -1,  0,  0 |
| -1,  2, -1, -1 |
|  0, -1,  2,  0 |
|  0, -1,  0,  2 |

atlas> ^roots*coroots=Cartan_matrix(rd)
Value: true  {this confirms that these are valid roots and 
coroots of the root system}