Chevalley Basis Details

The details for the construction of the Chevalley basis are as follows. Let $\mathrm{\𝔤}\mathit{ }$be a real, split semi-simple Lie algebra. Start with a basis $\left[H_i \, X_{\mathrm{\α}}\, X_{-\mathrm{\α}}\right]$for $𝔤$, where $H_i$, $i \=1\, 2\, ..\. \, r$is a basis for a Cartan subalgebra, and where $X_{\mathrm{\alpha }}\, X_{-\mathrm{\alpha }}$, $\mathrm{\α} \in {\mathrm{\Δ}}^{\+}$(the positive roots), gives a root space decomposition for $\mathrm{\𝔤}$. By definition of a split, semi-simple Lie algebra, the root vectors are all real. Let $B$ be the Killing form. Scale the vectors$X_{\mathrm{\α}}$such that $B\left(X_{\mathrm{\α}}\, X_{-\mathrm{\α}}\right) \= 1$and set $H_{\mathrm{\α}} \= \left[X_{\mathrm{\α}\,}{X}_{-\mathrm{\α}}\right]$. Scale the vectors $X_{\mathrm{\α}}$again (preserving $B\left(X_{\mathrm{\α}}\, X_{-\mathrm{\α}}\right) \= 1$) so that the structure equations

$\left[H_{\mathrm{\alpha }\,}{X}_{\mathrm{\α}}\right] \= 2 X_{\mathrm{\α}}\,$$\left[X_{\mathrm{\alpha }\,}{X}_{-\mathrm{\alpha }}\right]\=-H_{\mathrm{\α}} \, \left[H_{\mathrm{\alpha }\,}{X}_{-\mathrm{\α}}\right] \= -2 X_{-\mathrm{\α}}$

hold. Let ${\mathrm{\Δ}}_{}^0 \= \left[{\mathrm{\α}}_1\, {\mathrm{\α}}_2 \, ..\. \, {\mathrm{\α}}_r\right]$be the simple roots, and set

$h_i \= H_{{\mathrm{\α}}_i}\, x_{i } \=X_{{\mathrm{\α}}_i} \, y_i \= Y_{{\mathrm{\α}}_i}\, i \= 1\,2\, ..\. \,r$.

This fixes the $3 r$vectors $\left[h_1\, h_{2 }\, ..\. \, h_{r }\, x_1\, x_2\, ..\. \, x_r\, y_1\, y_2\, ..\. \, y_r\right]$ in the Chevalley basis $\mathrm{\ℬ}$. Write

 ${\mathrm{\Delta }}^{}{{}^c }_{}\= {\mathrm{\Δ}}_{}^{\+} - {\mathrm{\Δ}}_{}^0\= \left\{{\mathrm{\α}}_{r\+1}\, {\mathrm{\α}}_{r\+2}\, ..\.\, {\mathrm{\α}}_{\mathrm{\𝓁}\mathit{ }}\right\}$.

We need to make one final scaling of the vectors $X_{{\mathrm{\α}}_{}}$,$X_{-\mathrm{\α}}$for$\mathrm{\α} \in {\mathrm{\Δ}}_{}^c$. We calculate the structure constants $\left[X_{\mathrm{\α}} \, X_{\mathrm{\β}} \right] \= N_{\mathrm{\α} \mathrm{\β}} X_{\mathrm{\α} \+ \mathrm{\β}}$, for $\mathrm{\α}\, \mathrm{\β}$and $\mathrm{\α} \+ \mathrm{\β} \in {\mathrm{\Δ}}^{\+}$and generate the system of quadratic equations

${\left(q\+1\right)}^2{t}_{\mathrm{\α}} t_{\mathrm{\β}} \= N_{\mathrm{\α} \mathrm{\β} }^2 t_{\mathrm{\α} \+\mathrm{\β}}$ .

Here $q$is the largest positive integer such that $\mathrm{\α} -q \mathrm{\β}$is not a root. Put $t_{\mathrm{\α}} \= 1$for $\mathrm{\α} \in {\mathrm{\Δ}}_{}^0$and solve for the remaining $t_{\mathrm{\α}}\, \mathrm{\α} \in {\mathrm{\Δ}}^c$. Finally set $u_{\mathrm{\α} }$$\= \sqrt{t_{\mathrm{\α}}}$ and put$$

 $x_i\= u_{{\mathrm{\α}}_i} X_{{\mathrm{\α}}_i}$ and  $y_i \= 1\/u_{{\mathrm{\α}}_i} X_{-{\mathrm{\α}}_i}$for $i \= r\+1\, r\+2\, ..\. \mathrm{\𝓁}$.

This completes the construction of the Chevalley basis $\mathrm{\ℬ}$' $\left[h_1\, h_{2 }\, ..\. \, h_{r }\, x_1\, x_2\, ..\. \, x_{\mathrm{\ℓ}}\, y_1\, y_2\, ..\. \, y_{\mathrm{\ℓ}}\right]$. We have

$\left[h_i\, h_j\right] \= 0\, \left[ h_{i\,}{x}_j\right] \= a_{\mathrm{ij}}{x}_j \, \left[ h_{i\,}{y}_j\right] \= -a_{\mathrm{ij}}{x}_j$

for all $i\,j \= 1\,2\, ..\. r$, where the matrix $a_{\mathrm{ij}}$ is the Cartan matrix for $\mathrm{\𝔤}$ and, also,

$\left[x_i \, x_j\right] \=$$\pm \left(q\+1\right)x_k$ where ${\mathrm{\α}}_i \+ {\mathrm{\α}}_j \= {\mathrm{\α}}_k$.

Note that in the Chevalley basis all the structure constants are integers and that the transformation $h_{i } \to -h_i$, $x_i \to y_i\, y_i \to x_i$ is a Lie algebra automorphism.

See N. Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Section 4 for additional details.