Borel subalgebra

Let $\\mathfrak{g}$ be a semi-simple Lie group, $\\mathfrak{h}$ 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 $\\mathfrak{g}_{\\alpha}$

\\mathfrak{g}=\\mathfrak{h}\\oplus\\left(\\bigoplus_{\\alpha\\in R}\\mathfrak{g}_{%\\alpha}\\right).

Now let $\\mathfrak{b}$ be the direct sum of the Cartan subalgebra and the positive root spaces.

\\mathfrak{b}=\\mathfrak{h}\\oplus\\left(\\bigoplus_{\\beta\\in R^{+}}\\mathfrak{g}_{%\\beta}\\right).

This is called a Borel subalgebra.