Borel and radical

Question

I’m a little bit confused, could anyone explain what is the difference between the Borel subalgebra of $\mathfrak{g}$ and a radical of $\mathfrak{g}$?

Answer 1

The radical of $\mathfrak{g}$ is the largest solvable ideal, whereas a Borel subalgebra is a maximal solvable subalgebra. The solvable radical is unique, a Borel subalgebra need not be unique, and need not be an ideal.

For example, if $\mathfrak{g}=\mathfrak{sl}_n(\Bbb C)$, then ${\rm rad}(\mathfrak{g})=0$, whereas a Borel subalgebra is the Lie subalgebra of upper-triangular matrices of trace zero.