Root spaces of a complex semisimple Lie algebra are one-dimensional

Put $W=\mathbb{C}{e}_{\alpha }\oplus \mathbb{C}{h}_{\alpha }\oplus {⨁}_{k<0}{\mathfrak{g}}_{k\alpha }$, a finite-dimensional subspace of $\mathfrak{g}$ containing ${\mathfrak{g}}_{-\alpha }$. It is stable under ${\mathrm{ad}}_{h_{\alpha }}$ by [L2] together with $[h_{\alpha },e_{\alpha }]=2e_{\alpha }$ and $[h_{\alpha },h_{\alpha }]=0$ from [L1]; it is stable under ${\mathrm{ad}}_{e_{\alpha }}$ because $[e_{\alpha },{\mathfrak{g}}_{k\alpha }]\subseteq {\mathfrak{g}}_{(k+1)\alpha }$ with $(k+1)\alpha$ either $0$ or a negative multiple, $[e_{\alpha },{\mathfrak{g}}_{-\alpha }]\subseteq \mathbb{C}{H}_{\alpha }$ by [L3], and $[e_{\alpha },h_{\alpha }]=-2e_{\alpha }$; and it is stable under ${\mathrm{ad}}_{f_{\alpha }}$ because $[f_{\alpha },{\mathfrak{g}}_{k\alpha }]\subseteq {\mathfrak{g}}_{(k-1)\alpha }$ and $[f_{\alpha },h_{\alpha }]=2f_{\alpha }$, $[f_{\alpha },e_{\alpha }]=-h_{\alpha }$.

Since $h_{\alpha }=[e_{\alpha },f_{\alpha }]$, the restriction of ${\mathrm{ad}}_{h_{\alpha }}$ to the invariant subspace $W$ is a commutator of the restrictions of ${\mathrm{ad}}_{e_{\alpha }}$ and ${\mathrm{ad}}_{f_{\alpha }}$, so its trace vanishes by [L4].

On the other hand ${\mathrm{ad}}_{h_{\alpha }}$ acts on $\mathbb{C}{e}_{\alpha }$ by the scalar $2$, on $\mathbb{C}{h}_{\alpha }$ by $0$, and on the eigenspace ${\mathfrak{g}}_{k\alpha }$, $k<0$, by the scalar $k\alpha (h_{\alpha })=2k$; hence $0=\mathrm{tr}({\mathrm{ad}}_{h_{\alpha }}{\mathrm{\mid }}_W)=2-2{\sum }_{j\ge 1}j\dim {\mathfrak{g}}_{-j\alpha }$, that is, ${\sum }_{j\ge 1}j\dim {\mathfrak{g}}_{-j\alpha }=1$. As the summands are nonnegative integers, $\dim {\mathfrak{g}}_{-\alpha }=1$ and $\dim {\mathfrak{g}}_{-j\alpha }=0$ for $j\ge 2$.

The argument is symmetric in $\alpha$ and $-\alpha$: the triple $(f_{\alpha },e_{\alpha },-h_{\alpha })$ satisfies the same relations with $-\alpha$ in place of $\alpha$ by [L1], and all the facts [L2]–[L4] are unchanged. Applying step 3.1 with $-\alpha$ therefore gives $\dim {\mathfrak{g}}_{\alpha }=1$ and $\dim {\mathfrak{g}}_{j\alpha }=0$ for $j\ge 2$, which proves the statement.