Positive and negative nilpotent subalgebras and the Borel

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root set $\mathrm{\Phi }$, and let ${\mathrm{\Phi }}^{+}$ be a positive system of the root system with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ of simple roots (Positive systems and simple roots, The roots form a reduced crystallographic Euclidean root system). Write ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$ for the negative roots. Define ${\mathfrak{n}}^{+}={\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\mathfrak{g}}_{\alpha },{\mathfrak{n}}^{-}={\sum }_{\alpha \in {\mathrm{\Phi }}^{-}}{\mathfrak{g}}_{\alpha },\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}\mathrm{.}$ Then ${\mathfrak{n}}^{+}$ and ${\mathfrak{n}}^{-}$ are called the positive and negative nilpotent subalgebras and $\mathfrak{b}$ the Borel subalgebra attached to ${\mathrm{\Phi }}^{+}$. Here the term Borel uses this root-space construction as its defining convention, as in Knapp, Chapter V §7. The definition depends only on the set ${\mathrm{\Phi }}^{+}$ and not on any enumeration of it, because each sum is the span of a fixed set of subspaces.

These are Lie subalgebras. Let $\alpha ,\beta \in {\mathrm{\Phi }}^{+}$ and $x\in {\mathfrak{g}}_{\alpha }$, $y\in {\mathfrak{g}}_{\beta }$. By Brackets of root spaces, $[x,y]\in {\mathfrak{g}}_{\alpha +\beta }$, and ${\mathfrak{g}}_{\alpha +\beta }=0$ unless $\alpha +\beta$ is a root. When $\alpha +\beta$ is a root, write $\alpha ={\sum }_i{m}_i{\alpha }_i$ and $\beta ={\sum }_i{n}_i{\alpha }_i$ with nonnegative integral coefficients, as Simple roots form a signed integral basis permits; then $\alpha +\beta ={\sum }_i(m_i+n_i){\alpha }_i$ has nonnegative integral coefficients and is nonzero, so by the same theorem it is a positive root. Hence $[{\mathfrak{n}}^{+},{\mathfrak{n}}^{+}]\subseteq {\mathfrak{n}}^{+}$, and ${\mathfrak{n}}^{+}$ is a subalgebra; the same argument with signs reversed gives $[{\mathfrak{n}}^{-},{\mathfrak{n}}^{-}]\subseteq {\mathfrak{n}}^{-}$. Finally $[\mathfrak{h},\mathfrak{h}]=0$ and $[\mathfrak{h},{\mathfrak{g}}_{\alpha }]\subseteq {\mathfrak{g}}_{\alpha }$ for every root $\alpha$, so $[\mathfrak{h},{\mathfrak{n}}^{+}]\subseteq {\mathfrak{n}}^{+}$ and $\mathfrak{b}$ is closed under the bracket (Lie subalgebras, ideals, and center).

The subalgebras ${\mathfrak{n}}^{\pm }$ are nilpotent. For a positive root $\gamma ={\sum }_i{n}_i{\alpha }_i$ put $\mathrm{ht}(\gamma )={\sum }_i{n}_i$ and, for $k\ge 1$, $F_k=\mathrm{span}\{{\mathfrak{g}}_{\gamma }:\gamma \in {\mathrm{\Phi }}^{+},\mathrm{ht}(\gamma )\ge k\},$ the span being $0$ when no such root exists; by Root-space decomposition we have ${\mathfrak{n}}^{+}=F_1$, and $\mathrm{ht}(\gamma )\ge 1$ for every positive root because $\gamma \mathrm{\ne }0$ has nonnegative integral coefficients. If $\alpha ,\gamma \in {\mathrm{\Phi }}^{+}$ and $\alpha +\gamma$ is a root, then $\mathrm{ht}(\alpha +\gamma )=\mathrm{ht}(\alpha )+\mathrm{ht}(\gamma )$ by uniqueness of the simple-root coefficients, so $[{\mathfrak{n}}^{+},F_k]\subseteq F_{k+1}$. Induction gives ${\gamma }_k({\mathfrak{n}}^{+})\subseteq F_k$ for the lower central series (Lower central series and nilpotent Lie algebras). If ${\mathrm{\Phi }}^{+}$ is empty, then ${\mathfrak{n}}^{+}=0$ is nilpotent. Otherwise the finite nonempty set ${\mathrm{\Phi }}^{+}$ has a maximal height $H$, so $F_{H+1}=0$ and hence ${\gamma }_{H+1}({\mathfrak{n}}^{+})=0$: the algebra ${\mathfrak{n}}^{+}$ is nilpotent. The negative case is identical, with heights of the positive roots $-\gamma$ for $\gamma \in {\mathrm{\Phi }}^{-}$, since ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$. Thus the terms "positive and negative nilpotent subalgebras" are justified.