Let ${\displaystyle {\mathfrak {g}}={\mathfrak {gl}}(V)}$  be the Lie algebra of the endomorphisms of a finite-dimensional vector space V over the complex numbers. Then to specify a Borel subalgebra of ${\displaystyle {\mathfrak {g}}}$  amounts to specify a flag of V; given a flag ${\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0}$ , the subspace ${\displaystyle {\mathfrak {b}}=\{x\in {\mathfrak {g}}\mid x(V_{i})\subset V_{i},1\leq i\leq n\}}$  is a Borel subalgebra,^[2] and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of V.

Let ${\displaystyle {\mathfrak {g}}}$  be a complex semisimple Lie algebra, ${\displaystyle {\mathfrak {h}}}$  a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then ${\displaystyle {\mathfrak {g}}}$  has the decomposition ${\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$  where ${\displaystyle {\mathfrak {n}}^{\pm }=\sum _{\alpha >0}{\mathfrak {g}}_{\pm \alpha }}$ . Then ${\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$  is the Borel subalgebra relative to the above setup.^[3] (It is solvable since the derived algebra ${\displaystyle [{\mathfrak {b}},{\mathfrak {b}}]}$  is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.^[4])

Given a ${\displaystyle {\mathfrak {g}}}$ -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for ${\displaystyle {\mathfrak {h}}}$  and that (2) is annihilated by ${\displaystyle {\mathfrak {n}}^{+}}$ . It is the same thing as a ${\displaystyle {\mathfrak {b}}}$ -weight vector (Proof: if ${\displaystyle h\in {\mathfrak {h}}}$  and ${\displaystyle e\in {\mathfrak {n}}^{+}}$  with ${\displaystyle [h,e]=2e}$  and if ${\displaystyle {\mathfrak {b}}\cdot v}$  is a line, then ${\displaystyle 0=[h,e]\cdot v=2e\cdot v}$ .)

• Borel subgroup
• Parabolic Lie algebra