1. Definitions
For $V$ a vector space over $\mathbb C$ we call a subset $R \subset V$ an abstract root system if:
(1) The set $R$ is finite, spans $V$ and $0 \notin R$.
(2) For every $\alpha$ in $R$ there exists a linear map $s_\alpha:V \rightarrow V$ such that:
- $s_\alpha(\alpha)=-\alpha$
- $s_\alpha(R) \subset R$
- $s_\alpha(\beta) - \beta \in \mathbb{Z}\alpha$.
The subgroup $W \subset GL(V)$ generated by the reflections $s_\alpha$ is called the Weyl group.
One shows that given a complex semisimple Lie algebra $\mathfrak g$ and a Cartan subalgebra $\mathfrak{h \subset g}$ the roots $R(\mathfrak {g,h})$ of the corresponding root space decomposition form an abstract root system (with reflections $s_\alpha(\gamma)= \gamma - \alpha^{\vee} (\gamma)\alpha$ where $\alpha^{\vee} \in \mathfrak {h}$ is defined by the two conditions $\alpha^{\vee}(\alpha)=2$ and $\alpha^{\vee} \in [\mathfrak {g_\alpha, g_{-\alpha}}]$ with $g_\alpha$ the root spaces corresponding to the root $\alpha$).
2. Question
Consider the Lie algebra $\mathfrak {g:=so(2n, \mathbb{C})}$ with Cartan subalgebra $\mathfrak h := \{ \text{diagonal matrices in } \mathfrak g \}$.
I was able to show that the set $\{ \pm \epsilon_i \pm \epsilon_j:1 \leq i,j \leq n, i \neq j\}$ (with $\epsilon_i: \mathfrak h \rightarrow \mathbb C$ the linear map picking the i-th diagonal entry) forms a root system $R(\mathfrak {g,h})$. Now I want to determine the corresponding Weyl group. How could I do that with the definitions given? If I try to evaluate the roots on the reflections I do not know how to deal with $\alpha^{\vee}$.
