Let $G=\mathrm{GL}_{{n}}\mathbb{C}$, the group of all automorphisms of the $n$-dimensional vector space over the field of complex numbers $\mathbb{C}$, and $H\leq G$ 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 $K$.