Editing Maximal torus (section)

==Properties==
Let ''G'' be a compact, connected Lie group and let <math>\mathfrak g</math> be the [[Lie algebra]] of ''G''. The first main result is the torus theorem, which may be formulated as follows:<ref>{{harvnb|Hall|2015}} Lemma 11.12</ref>
:'''Torus theorem''':  If ''T'' is one fixed maximal torus in ''G'', then every element of ''G'' is conjugate to an element of ''T''. 
This theorem has the following consequences:
* All maximal tori in ''G'' are conjugate.<ref>{{harvnb|Hall|2015}} Theorem 11.9</ref> 
* All maximal tori have the same dimension, known as the ''rank'' of ''G''.
* A maximal torus in ''G'' is a maximal abelian subgroup, but the converse need not hold.<ref>{{harvnb|Hall|2015}} Theorem 11.36 and Exercise 11.5</ref>
* The maximal tori in ''G'' are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of <math>\mathfrak g</math><ref>{{harvnb|Hall|2015}} Proposition 11.7</ref> (cf. [[Cartan subalgebra]])
* Every element of ''G'' lies in some maximal torus; thus, the [[Exponential_map_(Lie_theory)|exponential map]] for ''G'' is surjective.
* If ''G'' has dimension ''n'' and rank ''r'' then ''n'' &minus; ''r'' is even.