Let $\mf{g}$ be a semi-simple Lie group, $\mf{h}$ a Cartan subalgebra, $R$ the associated root system, and $R^+\subset R$ a set of positive roots. We have a root decomposition into the Cartan subalgebra and the root spaces $\mf{g}_\alpha$ $$\mf{g}=\mf{h}\oplus\left(\bigoplus_{\alpha\in R}\mf{g}_\alpha\right).$$ Now let $\mf{b}$ be the direct sum of the Cartan subalgebra and the positive root spaces. $$\mf{b}=\mf{h}\oplus\left(\bigoplus_{\beta\in R^+}\mf{g}_\beta\right).$$ This is called a Borel subalgebra.