What is the root system and the Weyl group of the group spin$(2n)$?

Question

Reading on root systems and Weyl groups, unfortunately I am highly confused when it comes to the spin-groups (the two-fold universal cover of SO$(2n, \mathbb{C})$, realizable as a quotiënt in a Clifford algebra). Does anyone maybe know what the Weyl group is of the group spin$(2n, \mathbb{C}))$? Or what the root system is? (Then I might be able to deduce the Weyl-group!) Or maybe some reference where it is explained? Most books I've tried only deal with the group SO$(2n, \mathbb{C}))$.

Thanks!

Answer 1

If $\pi:G\to H$ is a covering of Lie groups (such as $\operatorname{Spin}(2n,\Bbb C)\to\operatorname{SO}(2n,\Bbb C)$), then the derivative at $1$ of $\pi$ gives an isomorphism $$\operatorname{Lie}(G)\cong\operatorname{Lie}(H).$$ Since the root system and the Weyl group are defined intrinsically in the Lie algebra, this says that $G$ and $H$ have the same root-system and Weyl group.