View source for Borel subgroup is conjugate-dense in connected algebraic group

{{conjugate-dense subgroup statement}}

==Statement==

===Statement over an algebraically closed field===

Suppose <math>G</math> is a [[connected algebraic group]] over an [[algebraically closed field]] <math>K</math> and <math>B</math> is a [[fact about::Borel subgroup;1| ]][[Borel subgroup]] of <math>G</math>. Then, <math>B</math> is a [[conjugate-dense subgroup]] of <math>G</math>, i.e., we can write:

<math>G = \bigcup_{g \in G} gBg^{-1}</math>

===Statement over an arbitrary field===

The statement as made over an algebraically closed field does not work over an arbitrary field, but we can make suitable modifications and make sure it does (can we?).

==Particular cases==

{| class="sortable" border="1"
! Theorem !! Value of <math>G</math> !! Value of <math>B</math>
|-
| [[Triangulability theorem]] || [[general linear group]] ||  subgroup of upper-triangular matrices.
|-
| [[Spectral theorem for unitary matrices]] (not sure if this really fits) || [[unitary group]] || subgroup of unitary diagonal matrices. 
|}