Borel subalgebra

Remarks

Example: in the special linear Lie algebra $\mathfrak{sl}_2(k)$ with basis $e, h, f$ and $[h, e] = 2e$, $[h, f] = -2f$, $[e, f] = h$, let $\mathfrak b = kh + ke$. It is a subalgebra with $[\mathfrak b, \mathfrak b] = ke$ and $[ke, ke] = 0$, so it is solvable. The only subalgebra properly containing it is $\mathfrak{sl}_2(k)$, which is not solvable, so $\mathfrak b$ is a Borel subalgebra.

Non-example: the line $ke$ is a solvable subalgebra, but it fails clause 2, since it lies in the solvable subalgebra $kh + ke$.

In $\mathfrak{gl}_n(k)$ the upper triangular matrices form a Borel subalgebra, and by Lie's theorem every Borel subalgebra is the stabilizer of a complete flag of subspaces of $k^n$. Every solvable subalgebra lies in some Borel subalgebra, by finite dimension. For a semisimple $\mathfrak g$ with a Cartan subalgebra $\mathfrak h$ and a set $\Phi^+$ of positive roots, $\mathfrak b = \mathfrak h \oplus \bigoplus_{\alpha \in \Phi^+} \mathfrak g_\alpha$ is a Borel subalgebra. Some texts define Borel subalgebras only for semisimple $\mathfrak g$, or over $\C$.