root vector

《Class Field Theory》 · 印刷第 170 页 Section 7.1 的约定、第 176–177 页 Section 7.3「Roots and Root Spaces」的约定、Definition 7.13 及 Definition 7.15（省略中间内容）

For a complex semisimple Lie algebra $\mathfrak g$, we will always assume we have chosen a compact real form $\mathfrak k$ of $\mathfrak g$, so that $\mathfrak g=\mathfrak k_{\mathbb C}$. […] For the rest of this chapter, we assume that we have chosen a compact real form $\mathfrak k$ of $\mathfrak g$ and a maximal commutative subalgebra $\mathfrak t$ of $\mathfrak k$, and we consider the Cartan subalgebra $\mathfrak h=\mathfrak t+i\mathfrak t$. We assume also that we have chosen an inner product on $\mathfrak g$ that is real on $\mathfrak k$ and invariant under the adjoint action of $K$ (Proposition 7.4). […] Definition 7.13. A nonzero element $\alpha$ of $\mathfrak h$ is a root (for $\mathfrak g$ relative to $\mathfrak h$) if there exists a nonzero $X\in\mathfrak g$ such that $[H,X]=\langle\alpha,H\rangle X$ for all $H$ in $\mathfrak h$. […] Definition 7.15. If $\alpha$ is a root, then the root space $\mathfrak g_\alpha$ is the space of all $X$ in $\mathfrak g$ for which $[H,X]=\langle\alpha,H\rangle X$ for all $H$ in $\mathfrak h$. A nonzero element of $\mathfrak g_\alpha$ is called a root vector for $\alpha$.