Root space

Let $\mathfrak g$ be a complex semisimple Lie algebra and $\mathfrak h\subset\mathfrak g$ a Cartan subalgebra. For $\alpha\in\mathfrak h^*$, the root space (more generally, the $\alpha$-weight space for $\mathrm{ad}|_{\mathfrak h}$) is

If $\alpha\neq 0$ and $\mathfrak g_\alpha\neq 0$, then $\alpha$ is a root and $\mathfrak g_\alpha$ is its root space.

These spaces interact well with the Lie bracket: if $X\in\mathfrak g_\alpha$ and $Y\in\mathfrak g_\beta$, then

so $[\mathfrak g_\alpha,\mathfrak g_\beta]\subseteq \mathfrak g_{\alpha+\beta}$ (interpreting $\mathfrak g_{\alpha+\beta}=0$ if $\alpha+\beta$ is not a weight). This is one of the structural inputs for the root space decomposition.

Concrete calculation (the $\mathfrak{sl}_2$ case). Let $\mathfrak g=\mathfrak{sl}_2(\mathbb C)$ (see standard sl2 example) with basis

Take $\mathfrak h=\mathbb C\cdot H$. Then

Define $\alpha\in\mathfrak h^*$ by $\alpha(H)=2$. Then $\mathfrak g_\alpha=\mathbb C\cdot E$ and $\mathfrak g_{-\alpha}=\mathbb C\cdot F$, giving the familiar decomposition