A standard set of generators for a semisimple Lie algebra $ L $ is defined as:
{${x_\alpha}, {y_\alpha}, {h_\alpha} $}
Where: $ x_\alpha \in L_\alpha, $ $ y_\alpha \in L_{-\alpha}, $ $ [{x_\alpha}{y_{\alpha}}]=h_\alpha $ $\alpha \in \Delta $
Wouldn't the standard set of generators require the $ x_\alpha $, $ y_\alpha $, and $ h_\alpha $ of all simple roots of base $\Delta $ if there are is than 1 simple root in $\Delta $ ?
Help is greatly appreciated
The set of generators is $\{x_\alpha,y_\alpha,h_\alpha: \alpha\in \Delta\}$. So it has $3|\Delta|$ elements. http://en.wikipedia.org/wiki/Root_system_of_a_semi-simple_Lie_algebra#Associated_semi-simple_Lie_algebra