Let $G=\GL n\C$ , the group of all automorphisms of the $n$ -dimensional vector space over the field of complex numbers $\mathbb{C}$ , and $H\le G$ a subgroup of $G$ . The standard Borel subgroup of $H$ is the subgroup of $H$ consisting of all upper triangular matrices (in $H$ ). A Borel subgroup of $H$ is a conjugate (in $H$ ) of the standard Borel subgroup of $H$ .

The notion of a Borel subgroup can be generalized. Let $G$ be a complex semi-simple Lie group. Then any maximal solvable subgroup $B\leq G$ is called a Borel subgroup. All Borel subgroups of a given group are conjugate. Any Borel group is connected and equal to its own normalizer, and contains a unique Cartan subgroup. The intersection of $B$ with a maximal compact subgroup $K$ of $G$ is the maximal torus of $K$ .